A Soup Can Is In The Shape Of A Cylinder? The 49 Latest Answer

We assume that beverage cans are cylindrical – the shape fits well in the hand and the cans can be stacked well on top of each other. But how did today’s can design become the standard? Finally, cylindrical cans don’t pack up as well as cube-shaped cans, and they use more metal than spherical cans.

A new video by University of Illinois chemical engineering professor Bill Hammack, who has a YouTube channel called Engineerguy, explains the science of how the modern soda can came to be.

A spherical can uses up the least amount of packaging, but would of course roll off the table – so out with it. A cube shaped can wouldn’t work as the edges are weak points and the walls would need to be made much thicker to withstand the pressure of the fizzy drink inside. (It’s also not particularly easy to hold or drink from.)

A cylindrical box combines the best properties of a sphere and a cube. Packed in a carton, bottles take up about 90 percent of the available space and their round shape withstands good pressurization. Modern aluminum cans are less than a tenth of a millimeter thick but hold liquids up to 90 pounds per square inch (about six times regular atmospheric pressure).

The aluminum or tin plated can begins life as a flat disc a few centimeters in diameter and is mechanically pressed into a flat cup shape and then into a taller cup that is the same diameter as the final can. The bottom of the cup is then pressed into a concave dome shape, allowing the can to withstand greater pressures than if it were flat. The entire process takes just one-seventh of a second, allowing a single machine to produce around 100 million cans over a six-month period.

Finally, the outside of the can is decorated and the inside is sprayed with a coating that prevents the soda from taking on a metallic taste. The still-open lid of the can is tapered, and once the can is filled with soda or juice, a separate machine immediately places the lid on the can and seals it to the body. Soda, explains Dr. Hammack, is pressurized with carbon dioxide, while juice is pressurized with nitrogen. This internal pressure allows the can to be relatively strong despite its thin walls – think how easy it is to crush an empty can by hand compared to how difficult it would be to do the same with an unopened can .

Get stories that

strengthen and uplift daily. By signing up, you agree to our privacy policy. Already a subscriber? Sign in to hide ads.

The modern soda can also includes a small tab that opens the top of the can without detaching. Today this feature is ubiquitous, but up until the 1970s cans had a pull tab that detached from the can and the beaches were often littered with discarded pull tabs.

Most of us deal with modern beverage cans on a daily basis, but it’s easy to forget that they are carefully designed and manufactured with an incredible level of precision. The beverage industry produces approximately 100 billion cans each year thanks to design that results in strong, reliable and efficient cans.

cylinder

A cylinder is a closed body with two parallel (usually circular) bases connected by a curved surface.

Try this: In the image below, drag the orange dot to change the dimensions of the cylinder.

base and side

A cylinder is a geometric body that is very common in everyday life, such as a soup can. If you take it apart, you’ll find that it has two ends called bases, usually circular in shape. The bases are always congruent and parallel to each other. If you were to “unroll” the cylinder, you would find that when flattened, the side is actually a rectangle. (See surface of a cylinder).

Height

The height h is the vertical distance between the bases. It is important to use the vertical height (or “height”) when calculating the volume of an inclined cylinder.

radius

The radius r of a cylinder is the radius of a base. If you get the diameter instead, remember to cut it in half.

axis

A line connecting the middle of each base.

Right and oblique cylinders

If the two bases are exactly on top of each other and the axis is at right angles to the base, it is called a “right cylinder”. When a base is shifted sideways, the axis is not perpendicular to the bases and the result is called an oblique cylinder. The bases, although not directly on top of each other, are still parallel.

In the applet at the top of the page, check the Allow Skew box and drag the orange dot sideways to reveal a skewed cylinder.

Right cylinder Oblique cylinder

volume and surface area

The shape of the bases

Usually the bases are circles, so a well-known soup can would technically be called a “right circular cylinder”. This is the most common way, and when someone just says “top hat,” that’s usually what they mean. However, the bases can be almost any curved shape, but the most common alternative to a circle is an ellipse. The shape would then be called an “elliptical cylinder”.

relation to a prism

A prism is a solid whose bases are polygons and the sides are flat. (See definition of a prism). Strictly speaking, a cylinder is not a prism, but it is very similar. For a prism where the bases are regular polygons, the prism approaches a cylinder when the number of sides is large.

See Relation of a Cylinder to a Prism for more information.

Another way to make a cylinder

Another way to create a circular cylinder is to think of it as the locus of a line moving parallel to and a fixed distance from the axis.

For more information, see Cylinder as a Line Location.

Related topics

(C) 2011 Copyright Math Open Reference.

All rights reserved

cylinder

Cylinder is one of the basic 3D shapes in geometry that has two spaced parallel circular bases. The two circular bases are connected by a curved surface at a fixed distance from the center. The line segment connecting the centers of two circular bases is the axis of the cylinder. The distance between the two circular bases is called the height of the cylinder. LPG gas cylinders are one of the real examples of cylinders.

Since the cylinder is a three-dimensional shape, it has two main properties, namely surface area and volume. The total surface area of ​​the cylinder is equal to the sum of its curved surface and the area of ​​its two circular bases. The space occupied by a cylinder in three dimensions is called its volume.

Here we will get acquainted with its definition, formulas and properties of the cylinder and solve some examples based on them. Apart from this figure, we have concepts of sphere, cone, box, cube, etc. that we learn in Solid Geometry.

definition

In mathematics, a cylinder is a three-dimensional solid holding two parallel bases connected by a curved surface at a fixed distance. These bases are usually circular (like a circle) and the center of the two bases is connected by a line segment called an axis. The perpendicular distance between the bases is the height “h” and the distance from the axis to the outer surface is the radius “r” of the cylinder.

Below is the illustration of the cylinder with area and height.

cylindrical shape

A cylinder is a three-dimensional shape composed of two parallel circular bases connected by a curved surface. The centers of the circular bases overlap to form a right cylinder. The line connecting the two centers is the axis denoting the height of the cylinder.

The top view of the cylinder looks like a circle and the side view of the cylinder looks like a rectangle.

Unlike cones, cubes, and parallelepipeds, a cylinder has no corners because the cylinder has a curved shape and no straight lines. It has two circular faces.

Characteristics

Each shape has some characteristics that distinguish one shape from another. Therefore, cylinders also have their properties.

The bases are always congruent and parallel.

If the axis forms a right angle with the bases lying exactly one on top of the other, then one speaks of a “right cylinder”.

It is similar to the prism in that it has the same cross-section everywhere.

If the bases are not exactly on top of each other, but sideways, and the axis does not form a right angle to the bases, then one speaks of an “oblique cylinder”.

If the bases are circular, it is called a right circular cylinder.

If the bases have an elliptical shape, then one speaks of an “elliptical cylinder”.

Find out more about the properties of the cylinder.

formulas

The cylinder has three main sizes, based on which we have the formulas.

Side face or curved face

total area

volume

Now let’s see their formulas.

Curved surface of the cylinder

The area of ​​the curved surface of the cylinder contained between the two parallel circular bases. It is also given as the lateral area. The formula for this is given by:

Curved surface area = 2πrh square units

total surface of the cylinder

The total surface area of ​​a cylinder is the sum of the curved surface area and the area of ​​two circular bases.

TSA = Curved Surface + Area of ​​Circular Bases

TSA = 2πrh + 2πr2

We can see from the above expression that 2πr is common. Because of this,

Total surface area, A = 2πr(r+h) square units

volume of the cylinder

Every three-dimensional shape or body has a volume that occupies some space. The volume of the cylinder is the space it occupies in any three-dimensional plane. The amount of water that could be immersed in a cylinder is described by its volume. The formula for the volume of the cylinder is:

Volume of the cylinder, V = πr 2 h cubic units

Where “h” is the height and “r” is the radius.

Also read: Area of ​​Hollow Cylinder

Solved examples

Question 1: Find the total surface of the cylinder with a radius of 5 cm and a height of 10 cm?

Solution: We know from the formula,

Total surface area of ​​a cylinder, A = 2πr(r+h) square units

Therefore A = 2π × 5(5 + 10) = 2π × 5(15) = 2π × 75 = 150 × 3.14 = 471 cm2

Question 2: What is the volume of a cylindrical water container with a height of 7 cm and a diameter of 10 cm?

Solution: given

Diameter of the container = 10 cm

Thus the radius of the container = 10/2 = 5 cm

Height of the container = 7cm

As we know from the formula,

Volume of a cylinder = πr2h cubic units.

Therefore the volume of the given container is V = π × 52 × 7

V = π × 25 × 7 = (22/7) × 25 × 7 = 22 × 25

V = 550 cm3

Nov 18, 2014 20:00:00

Why is the can not a sphere or a cube, but a cylinder?

This page contains information to support educators and families in teaching K-3 students about solid forms. It was designed to complement the Solid Shapes theme page on BrainPOP Jr.

Your children are most likely familiar with simple, two-dimensional planar shapes such as squares, rectangles, circles, and triangles. As a review, we recommend watching the movie Plane Shapes together. You may have experience with three-dimensional solids from block building, but may not have the vocabulary to describe and discuss them. You can build on your knowledge of body figures, which are three-dimensional shapes such as cubes, rectangular prisms, pyramids, cylinders, cones and spheres. Your children should be able to identify basic solid shapes and understand their similarities and differences. Begin by introducing basic concepts and vocabulary of geometry such as vertices, edges, faces, and bases. Encourage your kids to find basic solid shapes all around them.

Most children know dice. Number cubes or dice, ice cubes and some boxes are in the shape of dice. A cube has six flat surfaces or faces. Each face is shaped like a square with sides of equal length. A cube also has twelve edges and eight corners. Remind your kids that a vertex is a corner of a shape. The plural form of the word vertex is vertices. Present different examples of cubes to your children and help them identify the faces, edges and corners. A rectangular prism is a solid figure that has the same number of faces, edges, and vertices as a cube. As the name suggests, the faces of a rectangular prism are shaped like rectangles. Some rectangular prisms have faces shaped like rectangles and squares. Explain to your children that a cube is a special type of rectangular prism. Encourage your children to find examples of rectangular prisms and point out the different faces. Bulletin boards, cereal boxes, shoe boxes, and books are all rectangular prisms.

A pyramid is a solid figure that has a base or bottom and multiple triangular faces. The base of a pyramid can have different shapes, e.g. B. rectangles, squares, triangles or octagons. The sides join at a single vertex. A square pyramid has a base, four sides, eight edges, and five corners. Your children have most likely seen images of the Great Pyramids of Egypt. You may want to draw triangular and rectangular pyramids for your kids so they can see that no matter what shape the base is, the sides are always triangles.

A cylinder is a solid shape that has two circular faces, no edges, and no vertices. A cylinder has a curved surface and can roll. Tuna cans, soup cans, sticks, and whistles are examples of cylinders. You might want to show how a cylinder is made by rolling a rectangle into a tube and attaching two circles to the ends. Conversely, you can take a toilet paper roll and cut it lengthwise to show that it can turn into a rectangle. Help your kids understand that plane shapes can be manipulated to form solid shapes.

A cone is a solid shape that has a curved surface, no edges, and one vertex. Traffic cones, funnels, and ice cream cones are examples of cones. You can draw different examples of cones so students can see how they can vary in dimensions.

A sphere is a solid figure familiar to all children. Balls, marbles and oranges are all balls. A sphere has no face, edge, or vertex. Balls have a curved surface and can roll.

Studying shapes, both 2D and 3D, is a fun way for your kids to explore the world around them. How do we use rectangular prisms in everyday life? What form do we drink from? Encourage your children to find examples of solid shapes all around them and ask questions about how they use them. Would you rather play soccer with a ball or a box? Why?

Procedure 1.6. Drawing a soup can

Set the canvas parameters. Start Inkscape. Set the drawing size to 200 x 200 pixels. Set up a grid with a spacing of 5 pixels. Turn on general snapping (first ), node snapping (third , tooltip: snap to nodes, paths, and handles), smooth node snapping ( ), and grid snapping. Disable bounding box snapping (second , tooltip: Snap Bounding Box).

Draw the can shape. The box consists of two lines connecting parts of two ellipses. (One could use a rectangle to get the straight lines, but that has a tendency to create extra knots.) Draw the top of the can. Click on the Ellipse tool icon in the toolbox on the left of the Inkscape window (or use one of the keyboard shortcuts: F5 or e) to select the Ellipse tool. Draw an ellipse to represent the top of the can by clicking and dragging between the 50 and 150 pixel marks on the horizontal axis and between the 150 and 180 pixel marks on the vertical axis. Give the ellipse a solid fill by clicking on a colored square in the palette (I used light red). Draw the bottom of the can. Duplicate the ellipse by selecting the ellipse (if not selected) and clicking the icon on the command bar or using the Edit → Duplicate (Ctrl+D) menu item. A copy of the ellipse is placed over the original ellipse. The new ellipse remains selected. Move the ellipse down by clicking and dragging while holding down the Ctrl key to constrain movement to the vertical direction. Move it down until the peak is at the 50px mark on the vertical axis. You now have the top and bottom of the can. Draw the side of the can. The side of the can is formed by connecting the bottom half of the top ellipse to the bottom half of the bottom ellipse. In doing so, we sacrifice both ellipses. As we still need a separate top for the can as it will be colored differently than the body we will duplicate the top ellipse and sacrifice the new copy. Select the top ellipse and duplicate it as above. Then, while the double ellipse is still selected, convert the ellipse to a path object using the Path → Object to Path (Shift+Ctrl+C) command. The object doesn’t appear to have changed, but the underlying description is now an editable path. To edit the path, select the Node Tool by clicking the icon (F2 or n) in the toolbox. Select the top node of the path and delete it by clicking the (Delete Selected Nodes) icon in the Tool Control or by using one of the keyboard shortcuts: Backspace or Delete. Open the path by selecting the two side nodes and clicking the icon (Delete segment between two non-endpoint nodes) in the tool controls. Repeat the previous steps for the bottom ellipse (except for duplicating). You should have a drawing that looks like this if you change the color of the top ellipse, which is still intact: Next, the top and bottom ellipses need to be merged. Select both and combine them into one path using the Path → Combine (Ctrl+K) command. Using the node tool, select the two leftmost nodes, one from the top and one from the bottom. Connect them by clicking the (Connect selected end nodes to a new segment) icon in the tool controls. Repeat for the two rightmost nodes. You should now have a well constructed can side as shown below.

Add a gradient for a 3D effect. A gradient can represent the reflections of the curved part of the can. There are two ways to add a gradient, the first is by using the gradient tool and the second is by using the fill and stroke dialog. To add a gradient using the gradient tool, select the tool by clicking the icon (Ctrl+F1 or g) in the toolbox. Then, with the side of the can selected, click and drag from the left side of the can to the right side. To add a gradient using the Fill and Stroke dialog, open the dialog (Object → Fill and Stroke… (Shift+Ctrl+F)). Select the Fill tab if not already selected, and with the can side selected, click the Linear Gradient icon ( ). Both methods create a default gradient across the can side with the fill color already in place. The gradient requires a bit of work to make it look right, which is easiest to do with the gradient tool. Two gradient handles appear when the can side is selected. Gradients are defined in terms of stops. A stop has a color and position (offset) in the gradient. The default gradient has two stops, both the same color but with different transparencies. For the side of the can we use three stops. Add a third stop by double-clicking on the line connecting the two existing stops with the Gradient Tool active and the side of the can selected. The cursor has an additional + sign when it is possible to add stops. When you add a stop, it takes on the color of the gradient where it’s added. The appearance of the gradient does not change. The stop (handle) can be dragged to move it. Move it to the center. Now let’s give our tin a shiny metallic look. Select the leftmost stop by clicking on it once. Change the color to dark gray either by clicking the 80% Gray swatch in the palette (a tooltip with the color name appears when the mouse pointer is over a swatch), or by using the Fill and Stroke dialog box. Both the Fill and Stroke tabs show the color of the stop and can be used to change the color. With the RGB tab; Change the values ​​to R: 51, G: 51, B: 51, A: 255. Select the middle stop and set its color to white with the palette or with R: 255, G: 255, B: 255, A: 255 on the Fill and Stroke dialog box. Finally, select the last stop and set its color to match the first stop. You should now have a metallic can with a highlight in the middle. Let’s move the highlight to the side. One way to do this is to further edit the gradient, changing the position of the center stop, and maybe lightening or darkening the side stops. However, the easier way is to move the gradient handles. Before we continue, let’s make a few quick cosmetic changes: Disable the stroke on both the can side and the top (click the x at the left end of the palette while holding down Shift). Change the Top Color to 40% Gray (R:153, G:153, B:153).

Adding a shadow. A light shines on our can from the right, but no corresponding shadow. We’re going to fix that now. Making it perfect is not an easy task. Inkscape is a 2D drawing program and cannot project shadows for 3D objects (try POV-Ray for that). But we can make a pretty good approximation. Create the shadow object. For the shadow we need to combine copies of the top and side of the can into one object. We’ll play a game similar to the one used to create the side of the can. Select the top of the can (an ellipse) and duplicate it. Change its color to make it easier to follow. Any color will do. Convert the new ellipse to a path (Path → Object to Path (Shift+Ctrl+C)). Remove the bottom knot (switch to the knot tool, ). Select the two side nodes and remove the line between them ( ). Select the side of the can, duplicate it and change its color. Remove the top three knots. Select the remaining two side nodes and remove the line between them. Select both new objects. Combine them into a path (Path → Combine (Ctrl+K)). Select the two nodes on the left, connect them ( ) and convert the path between them to a line ( ). Do the same for the two right nodes. Distort the shadow shape. Select the top three nodes and move them down and up as shown below. It is clear that the side nodes are not positioned correctly and the path is distorted. We’re going to make a few adjustments to fix this. But first place the shadow object behind the can (Object → Lower to Bottom (End)). Node grips are used to control the direction and curvature of a path on either side of a node. The handles are the circles attached to a node by straight lines that appear when the node is selected. The lines are always tangent to the path at the point where the path intersects the node. The distance between the handle and its node controls the curvature of the path—the farther away, the less curved the path near the node. If no node handles are visible, make them visible by clicking the icon in the tool controls, or if the icon is not visible, check the Show handles box in the drop-down menu opened by clicking the down arrow at the right end becomes the tool control. The nodes indicated by the arrows in the image below need to be adjusted. The handle on the left node needs to be rotated counterclockwise a little so that the handle line is parallel to the side of the shadow. This creates a smooth transition between the straight side and the curved part of the shadow. To make the rotation, drag the handle while holding down the Alt key (which keeps the distance between the handle and the node constant). The handle of the other node, indicated by the arrow, also needs to be adjusted in the same way. Also, the node should be moved slightly to the lower right so that the straight edge of the shadow is tangent to the bottom of the can. Now change the color of the fill to 50% Gray (R:127 G:127 B:127). Soften the edge of the shadow. The easiest way to soften the shadow edge is to use the Blur slider on the Fill tab of the Fill and Stroke dialog. A value of 3% gives a nice shadow. Select the new blurred shadow. Move the shadow to the back (Object → Bottom to Bottom (End)). A problem needs to be solved. The shadow exits the bottom of the can at the front. Select the shadow and slightly move it up and to the left. You can do this by either dragging the shadow with the mouse or by using the arrow keys. Holding down the Alt key while using the arrow keys allows you to make finer adjustments.

Add Label. The label has the same curvature as the sidewall of the can. To create the label, we’ll take a piece of the side of the can and then lower its height. Select the side of the can and duplicate it. Then select the Rectangle Tool and draw a rectangle that extends from above the can to below the can and stretches from 60 to 140 pixels in the horizontal direction. Select both the new rectangle and the duplicate can side. Use the Path → Intersection (Ctrl+*) command to form the label from the intersection of the two selected objects. Change the label’s fill to a solid red by clicking the red square in the palette, or first clicking the flat color icon ( ) on the Fill tab of the Fill and Stroke dialog box, and then clicking the sliders set red (R: 255 :G:0:B:0). Change the opacity to 50% using the 0: edit box in the style viewer (in the lower-left corner of the Inkscape window) or the Opacity slider in the Fill and Stroke dialog. This may cause some of the can to protrude. (If you want a metallic label, just change the color of the gradient.) Using the node tool, simultaneously select the three nodes at the top of the label and move them down 25px. Move the bottom three nodes of the label 25 pixels up.

Add text to the label. To add text to the label, select the Text tool. Click anywhere on the canvas and type the text “SPLIT PEA SOUP” with a carriage return after each word. To change the size and style of the text, use the drop down menus in the tool controls. Choose an appropriate font (I used Bitstream Vera Sans) and font size (18 pt or 24 px). The line spacing may have to be adjusted using the line spacing input field ( ). Change the line spacing to 1 unitless in the adjacent units drop-down menu. Finally, use the Fill and Stroke dialog to change the text to a nice pea green (R:63, G:127:B:0). Now align the text to the center of the can. First click the icon in the tool control to use the center of the text for snapping. Then, in the Snap bar, enable snapping to text using the Snap to other points (fourth ) and Anchor and baseline snap ( ) and grid snap ( ) icons. Finally, drag the text until it snaps to the center grid line. Now we have flat text on a round can! Inkscape can move and rotate individual text letters, but it cannot distort the text like the characters to the edge of the can do. To distort the letters, we’ll convert the text to a path and change the path. To do this, select the text and then use the Path → Object to Path (Shift+Ctrl+C) command. The text has become a group of path objects and can no longer be edited as text. It’s easier to change the paths if they just form one path. Group the objects with Object → Ungroup (Shift+Ctrl+G) and then combine the individual letters into a path with Path → Combine (Ctrl+K). We’re going to use a couple of very handy extensions to do the hard work for us. Start with the Add Nodes expansion to increase the number of nodes in the path. This improves the appearance of the last few letters. Select the text path and then choose Extensions → Change Path → Add Node… . A small dialog will appear in which you can set the upper limit for the distance between the nodes. Set the maximum segment length to 5.0 and click the Apply button. Next, we’ll use the Pattern Along Path extension to place the text on a path. There is a Text → Put on Path command, but this one uses the SVG textPath specification, which would result in the letters being rotated to follow the path. We need a path to place the text. The path should be an arc with the same shape as the curve of the top ellipse but the width of the text. Select the top ellipse on top of the can and duplicate it (Edit → Duplicate (Ctrl+D)). Move the ellipse straight down to the center of the can. Remove the fill but add a stroke. Add a rectangle overlapping the bottom of the ellipse, centered on the can and the width of the text. The rectangle should be above the ellipse. Select both and use the Path → Cut Path command (Ctrl+Alt+/) to split the ellipse in two. The rectangle disappears. Use the node tool to select the top piece of the ellipse, then delete it (Edit → Delete (Delete or Backspace)). The expansion requires that the pattern (text in this case) be above the path in z-order (i.e. above the path). You can move the pattern up by selecting it and then clicking the Raise Up icon (Object → Raise Up (home)). Next, select the text and the path. Call the Extension Pattern Along Path (Extensions → Generate from Path → Pattern Along Path). This opens a dialog. In the dialog, select Individual from the Copies of Pattern menu and Band from the Deformation Type menu. All input fields should be 0,0 and none of the boxes should be checked. Click the Apply button. The text might be backward. In this case, rotate it using the Object → Flip Horizontal (H) command. Finally, erase the small arc.

Activity: soup can

Pi Soups Incorporated wants to sell soup in cylindrical cans containing 400 ml (milliliter).

You also want to minimize the amount of metal in each can.

As a local math expert, it’s your job to help them!

volume and surface area

For a cylinder of height h and base radius r: surface area = 2 × π × r × (r + h)

× r × (r + h) Volume = π × r2 × h Which can be abbreviated to: A = 2 π r(r + h)

r(r + h) V = π r2h

thickness of metal

Let’s assume the metal is a uniform thickness, so let’s just find out

The smallest surface that holds 400ml

Also assume that the thickness of the metal does not affect the volume (i.e. it is very small compared to the size of the can).

To solve this we will work in centimeters, so:

the area is measured in square centimeters (cm 2 ) and

) and the volume is measured in cubic centimeters (cm3).

That makes it easy, because 1 ml corresponds to a volume of 1 cm3 – see page Metric Volume.

So V = 400

This gives us the formula 400 = πr2h

Make h the subject of the formula:

Divide both sides by π r2: 400/ π r2 = h

Swap sides: h = 400/ π r2

Substitute this value for h into the formula for A:

A = 2πr(r+h) = 2πr(r + 400/πr2) = 2πr2 + 800/r

Now we know the area in terms of radius:

A = 2πr2 + 800/r

This is our very simple mathematical model of the soup can.

It’s your turn!

I’ve helped you enough. Now it is your turn.

You can use the above formula to study how changing the radius affects the area and find out which radius gives the least metal area.

You can create a report that explains things to everyone (imagine having to present it to an executive meeting) with a graph, a table, and words.

Graph: draw the graph of the function, find the minimum value (you can use the function graph)

Table: Find sample points on the graph by substituting different values ​​of r. For example, if r = 2, then A = 2π × 22 + 800/2 = 8π + 400 = 425 to the nearest integer. In other words, if the radius of the can is 2 cm, the metal area needed to make the can is 425 cm2. Write the example values ​​in a table like this: Radius Area Comments 0 1 2 3 4 5 6 7

Report: Compile the graph and table into a report with: an introduction,

an explanation

and your conclusions and recommendations.

A 1 liter can

What is the best radius for a 1 liter can of soup?

improvement of the model

air gap

Soup cans are not completely filled with soup…there is a small air gap at the top. You can model this by adding (let’s say) 1 cm to the height.

Can you write the formula including this change?

How does this affect the chart and minimum value?

rims

Soup cans have extra thick rims at the top and bottom, how could you factor that into the formula?

Circular cylinder shape

r = radius

h = height

V = volume

L = side face

T = top surface

B = base area

A = total area

π = pi = 3.1415926535898

√ = square root

use calculator

This online calculator calculates the various properties of a cylinder using 2 known values. It also calculates these properties in terms of PI π. This is a right circular cylinder where the top and bottom are parallel, but it is commonly referred to as a “top hat”.

Units: Note that the units are shown for convenience but do not affect the calculations. The units are there to show the order of the results, e.g. ft, ft2 or ft3. For example, if you start with mm and know r and h in mm, your calculations will give V in mm3, L in mm2, T in mm2, B in mm2, and A in mm2.

Below are the standard formulas for a cylinder. Calculations are based on algebraic manipulation of these standard formulas.

Cylinder formulas related to r and h:

Calculate the volume of a cylinder: V = π r 2 h

Calculate the surface area of ​​a cylinder (only the curved outside)**: L = 2 π rh

Calculate the top and bottom surfaces of a cylinder (2 circles): T = B = π r 2

The total surface area of ​​a closed cylinder is: A = L + T + B = 2π rh + 2( π r 2 ) = 2 π r(h+r)

** The calculated area is only the surface area of ​​the outer cylinder wall. To calculate the total area, you also need to calculate the area of ​​the top and bottom. You can do this with the circle calculator.

Cylinder Calculations:

Use the following additional formulas along with the formulas above.

The label of the soup is the side area of ​​the cylinder.

The formula for a cylinder is LA = ph.

The circumference of a cylinder is the circumference of a circle, 2 times pi times r.

then multiply that answer by the amount of 6.

The answer will be in square inches.

12 oz of liquid fit in a 355 ml can. Slim cans hold the same amount of liquid as their larger counterparts despite being tall and slim. Cans that are tall and slender typically measure 6 inches tall. There are 125 inches in height and two feet in width. Cans are generally 4″ in diameter, while 25″ cans are generally 25″ in diameter. I am 83 cm tall and weigh 2.2 kg. The diameter of the circle is 6 inches.

How tall is a normal beer can? In the US, the standard can size is 11 x 8 x 4 inches. is 12 fl. ounces. There are two cans in this picture. A diameter of 12 inches and a height of 4 inches. This event lasted about eight weeks.

How tall is a 16 ounce beer can? MATERIAL Aluminum VOLUME 16 oz / 473 ml BODY DIAMETER 211 FINISHED DIAMETER 202 FINISHED CAN HEIGHT 6.190” +/- 0.012”

What are the sizes of beer cans? A 12-ounce can is typically packaged in six, twelve, fifteen, eighteen, twenty, and thirty cans. In recent years, craft breweries have started using this size of can to showcase their most interesting creations. 16 ounces. Can: This is the same size as a pint and is growing in popularity.

What are the tall beer cans called? Despite the popularity of 12-ounce cans, craft brewers are beginning to use 16-ounce pint cans, also known as tallboys. The most common packaging for these items is a 4-pack.

How much beer is in a tall can? Can of Tallboy beer with a capacity of 710 ml, 24 US fl oz.

How many inches is a beer can tall? Aluminum cans known as “Brite” are unlabeled and range in size from 5 to 4 gallons. Five to 32 ounces in size. In the US, the standard can size is 11 x 8 x 4 inches. is 12 fl. ounces. There are two cans in this picture. A diameter of 12 inches and a height of 4 inches. This event lasted about eight weeks.

How many milliliters are in a large can of beer? There is no standard size for big boys, but they are a large can, with a volume of around 500ml.

How high are high doses? Cans that are tall and slender typically measure 6 inches tall. There are 125 inches in height and two feet in width. Cans are generally 4″ in diameter, while 25″ cans are generally 25″ in diameter. I am 83 cm tall and weigh 2.2 kg. The diameter of the circle is 6 inches.

How tall is a 22 ounce beer can? 22oz Bomber – Bomber. Something about this bottle makes it seem like the granddaddy of long-necked bottles.

How tall is a 32 ounce beer can? The aluminum body is available in 307mm diameters, 32oz (946mL) volumes, and 300mm and 710mm sizes. You can already order one pallet plus additional layers.

cylinder

Cylinder is one of the basic 3D shapes in geometry that has two spaced parallel circular bases. The two circular bases are connected by a curved surface at a fixed distance from the center. The line segment connecting the centers of two circular bases is the axis of the cylinder. The distance between the two circular bases is called the height of the cylinder. LPG gas cylinders are one of the real examples of cylinders.

Since the cylinder is a three-dimensional shape, it has two main properties, namely surface area and volume. The total surface area of ​​the cylinder is equal to the sum of its curved surface and the area of ​​its two circular bases. The space occupied by a cylinder in three dimensions is called its volume.

Here we will get acquainted with its definition, formulas and properties of the cylinder and solve some examples based on them. Apart from this figure, we have concepts of sphere, cone, box, cube, etc. that we learn in Solid Geometry.

definition

In mathematics, a cylinder is a three-dimensional solid holding two parallel bases connected by a curved surface at a fixed distance. These bases are usually circular (like a circle) and the center of the two bases is connected by a line segment called an axis. The perpendicular distance between the bases is the height “h” and the distance from the axis to the outer surface is the radius “r” of the cylinder.

Below is the illustration of the cylinder with area and height.

cylindrical shape

A cylinder is a three-dimensional shape composed of two parallel circular bases connected by a curved surface. The centers of the circular bases overlap to form a right cylinder. The line connecting the two centers is the axis denoting the height of the cylinder.

The top view of the cylinder looks like a circle and the side view of the cylinder looks like a rectangle.

Unlike cones, cubes, and parallelepipeds, a cylinder has no corners because the cylinder has a curved shape and no straight lines. It has two circular faces.

Characteristics

Each shape has some characteristics that distinguish one shape from another. Therefore, cylinders also have their properties.

The bases are always congruent and parallel.

If the axis forms a right angle with the bases lying exactly one on top of the other, then one speaks of a “right cylinder”.

It is similar to the prism in that it has the same cross-section everywhere.

If the bases are not exactly on top of each other, but sideways, and the axis does not form a right angle to the bases, then one speaks of an “oblique cylinder”.

If the bases are circular, it is called a right circular cylinder.

If the bases have an elliptical shape, then one speaks of an “elliptical cylinder”.

Find out more about the properties of the cylinder.

formulas

The cylinder has three main sizes, based on which we have the formulas.

Side face or curved face

total area

volume

Now let’s see their formulas.

Curved surface of the cylinder

The area of ​​the curved surface of the cylinder contained between the two parallel circular bases. It is also given as the lateral area. The formula for this is given by:

Curved surface area = 2πrh square units

total surface of the cylinder

The total surface area of ​​a cylinder is the sum of the curved surface area and the area of ​​two circular bases.

TSA = Curved Surface + Area of ​​Circular Bases

TSA = 2πrh + 2πr2

We can see from the above expression that 2πr is common. Because of this,

Total surface area, A = 2πr(r+h) square units

volume of the cylinder

Every three-dimensional shape or body has a volume that occupies some space. The volume of the cylinder is the space it occupies in any three-dimensional plane. The amount of water that could be immersed in a cylinder is described by its volume. The formula for the volume of the cylinder is:

Volume of the cylinder, V = πr 2 h cubic units

Where “h” is the height and “r” is the radius.

Also read: Area of ​​Hollow Cylinder

Solved examples

Question 1: Find the total surface of the cylinder with a radius of 5 cm and a height of 10 cm?

Solution: We know from the formula,

Total surface area of ​​a cylinder, A = 2πr(r+h) square units

Therefore A = 2π × 5(5 + 10) = 2π × 5(15) = 2π × 75 = 150 × 3.14 = 471 cm2

Question 2: What is the volume of a cylindrical water container with a height of 7 cm and a diameter of 10 cm?

Solution: given

Diameter of the container = 10 cm

Thus the radius of the container = 10/2 = 5 cm

Height of the container = 7cm

As we know from the formula,

Volume of a cylinder = πr2h cubic units.

Therefore the volume of the given container is V = π × 52 × 7

V = π × 25 × 7 = (22/7) × 25 × 7 = 22 × 25

V = 550 cm3

Circular cylinder shape

r = radius

h = height

V = volume

L = side face

T = top surface

B = base area

A = total area

π = pi = 3.1415926535898

√ = square root

use calculator

This online calculator calculates the various properties of a cylinder using 2 known values. It also calculates these properties in terms of PI π. This is a right circular cylinder where the top and bottom are parallel, but it is commonly referred to as a “top hat”.

Units: Note that the units are shown for convenience but do not affect the calculations. The units are there to show the order of the results, e.g. ft, ft2 or ft3. For example, if you start with mm and know r and h in mm, your calculations will give V in mm3, L in mm2, T in mm2, B in mm2, and A in mm2.

Below are the standard formulas for a cylinder. Calculations are based on algebraic manipulation of these standard formulas.

Cylinder formulas related to r and h:

Calculate the volume of a cylinder: V = π r 2 h

Calculate the surface area of ​​a cylinder (only the curved outside)**: L = 2 π rh

Calculate the top and bottom surfaces of a cylinder (2 circles): T = B = π r 2

The total surface area of ​​a closed cylinder is: A = L + T + B = 2π rh + 2( π r 2 ) = 2 π r(h+r)

** The calculated area is only the surface area of ​​the outer cylinder wall. To calculate the total area, you also need to calculate the area of ​​the top and bottom. You can do this with the circle calculator.

Cylinder Calculations:

Use the following additional formulas along with the formulas above.

volume of a cylinder

A cylinder is a solid composed of two congruent circles in parallel planes, their interiors, and any line segments that are parallel to the segment containing the centers of both circles with endpoints on the circular areas.

The volume of a three-dimensional solid is the amount of space it occupies. Volume is measured in cubic units (in 3 , ft 3 , cm 3 , m 3 , etc.). Make sure all measurements are in the same units before calculating volume.

The volume V of a cylinder of radius r is the area of ​​the base B times the height h.

V = B h or V = π r 2 h

surface of a cylinder

The surface area of ​​a cylinder is the area its surface occupies in three-dimensional space. A cylinder is a three-dimensional structure with circular bases that are parallel to each other. It has no corners. In general, the area of ​​three-dimensional shapes refers to the surface area. The area is represented in square units. For example cm2, m2 and so on. A cylinder can be thought of as a series of circular disks stacked on top of each other. Because the cylinder is a solid with a three-dimensional shape, it has both surface area and volume.

The cylinder area is defined as the sum of the curved surface and the area of ​​two circular bases of the cylinder.

The Surface Area of ​​the Cylinder = Curved Surface Area + Area of ​​Circular Base S.A. (in terms of π) = 2πr (h + r) square unit Where π (pi) = 3.142 or = 22/7 r = radius of the cylinder h = height of the cylinder

As we know, a cylinder has two types of surfaces, one is the curved surface and the other is the circular base. So the total surface area is the sum of the curved surface area and two circular bases.

Also read: Volume of Cylinder.

Table of Contents:

Cylinder area definition

The area of ​​the cylinder is the total area covered by a cylinder in three-dimensional space. The cylinder area is equal to the sum of the area of ​​two circular base areas and the curved lateral area. In straight cylinders, the two circular bases lie exactly on top of each other and the axis line forms a right angle to the base. If one of the circular bases is shifted and the axis does not form a right angle to the base, it is called an oblique cylinder.

In the center of the two round bases is a curved surface that when opened reveals a rectangular shape. This curved surface is also referred to as the lateral surface. The various parameters used to calculate the cylinder area include radius, height, axis, base, and side. The radius of the cylinder is defined as the radius of the circular base. The height of the cylinder is calculated by measuring the perpendicular distance between two circular bases, and the line connecting the center of the base is called the axis.

Area of ​​a cylinder formula

The total area of ​​a cylinder is given by:

Curved Surface (CSA)

Floor space

curved surface

The curved surface of a cylinder (CSA) is defined as the area of ​​the curved surface of any cylinder with a base radius “r” and a height “h”. It is also referred to as the lateral surface (LSA). The formula for a curved surface or lateral surface is given by:

CSA or LSA = 2π × r × h Square units

base of the cylinder

The base of the cylinder is circular. Therefore, by the area formula of the circle, we know

Area of ​​circular bases of cylinder = 2(πr2) [Since the cylinder has two circular bases]

total surface of the cylinder

The total area of ​​a cylinder is equal to the sum of the areas of all its areas. The total surface area of ​​radius “r” and height “h” is equal to the sum of the curved area and the circular area of ​​the cylinder.

TSA = 2π × r × h + 2πr2= 2πr (h + r) Quadratic units

Also read:

Derivation of the surface of the cylinder

Now imagine a scenario where we need to paint the faces of a cylindrical container. Before we start painting, we need to know the amount of paint that will be needed to paint all of the walls. Therefore, we need to find the area of ​​all faces of this container to calculate the amount of paint needed. We define this term as total area.

Consider a cylinder with units of base radius “r” and height “h”. The curved surface of this cylinder, when opened along the diameter (d = 2r) of the circular base, can be transformed into a rectangle of length “2πr” and width “h”.

From the formula for the area of ​​a circle, we know

Area of ​​the circular base of the cylinder = πr2

Since there are two circular bases, therefore the area of ​​the two circular bases = πr2 + πr2 = 2πr2 ……………….(1)

Now from the figure you can see that if we open the curved surface of the cylinder in two-dimensional space, it forms a rectangle. Thus, the height and perimeter of the circular bases are the dimensions of the rectangle formed from them. Because of this,

Area of ​​curved surface = height × perimeter

Curved surface = h × πd = h × 2πr (since d = 2r)

CSA = 2πrh …………….(2)

Adding Equation 1 and Equation 2 we get the total surface area such that;

Total area = curved area + area of ​​circular bases

TSA = 2πrh + 2πr2

Taking 2πr as the common factor of RHS we get;

TSA = 2πr (h + r)

This is the formula for the total surface area of ​​a given cylinder whose radius is r and whose height is h.

problems and solutions

Q.1: Calculate the cost of painting a container in the shape of a right circular cylinder with a base radius of 7m and a height of 13m. If the cost of painting the container is INR 2.5/m2. (Take π = 22/7)

Solution:

Total aquarium surface = 2πr (h + r)= 2 × (22/7) × 7 × 20 = 880 m2

Total cost of painting the tank = 2.5 × 880 = Rs. 2200

Q.2: Find the total surface area of ​​a container in a cylindrical shape with a diameter of 28 cm and a height of 15 cm.

Solution:

Given, diameter = 28 cm, so radius = 28/2 = 14 cm

and height = 15 cm

By the formula of the total area we are known;

TSA = 2πr (h + r) = 2 × (22/7) × 14 × (15 + 14)

TSA = 2x22x2x29

TSA = 2552 cm²

Thus the total surface area of ​​the container is 2552 cm².

practice questions

A water tank has a radius of 40 inches and a height of 150 inches. Find the area. Find the radius of a cylinder with a total surface area of ​​2136.56 square centimeters and a height of 3 cm. Find the curved surface of a cylinder with a diameter of 56 cm and a height of 20 cm.

To learn and practice more problems related to calculating the surface area and volume of a cylinder, download BYJU’S – The Learning App.