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Surface area and volume of a cylinder

Part of MathsPerimeter, Area, Volume

Key points

An image of an upright cylinder. Drawn right: an arrow to represent the height of the cylinder. The arrow is labelled as height. The top circular face of the cylinder is coloured orange and labelled: area of cross section. Two further circles through the shape, one being the base, have been coloured orange. The edges of these circles are drawn with a dashed curve. Downward orange arrows to illustrate the uniform cross section are drawn between the circles. Written below, the formula: volume equals area of cross section multiplied by height. The cylinder is coloured blue.
The volume of a cylinder is the area of the cross-section multiplied by the height.

A good understanding of calculations for the circumference of a circle and the area of a circle is useful when calculating the surface area and volume of a .

  • A cylinder is a shape with a circular .

  • The total of a cylinder is made up of two circular and a curved surface which makes a rectangle if flattened out. Surface area is measured in square units, such as cm虏 and m虏.

  • The of a cylinder is the of its cross-section, a circle, multiplied by its height. Volume is measured in cubic units, such as cm鲁 or m鲁.

An image of an upright cylinder. Drawn right: an arrow to represent the height of the cylinder. The arrow is labelled as height. The top circular face of the cylinder is coloured orange and labelled: area of cross section. Two further circles through the shape, one being the base, have been coloured orange. The edges of these circles are drawn with a dashed curve. Downward orange arrows to illustrate the uniform cross section are drawn between the circles. Written below, the formula: volume equals area of cross section multiplied by height. The cylinder is coloured blue.
The volume of a cylinder is the area of the cross-section multiplied by the height.
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Calculate the area of a cylinder

  • A cylinder is made up of two circles that are directly opposite one another, and a rectangle.

  • Calculations may be carried out numerically using a decimal approximation for (pi), such as 3郯14 or 3郯142. Workings may also be written symbolically in terms of 蟺. This means that the result is given as a multiple of 蟺.

  • On a scientific calculator, the S

    鈬

    D button is used to convert a value in terms of 蟺 to a decimal value.

To calculate the surface area of a cylinder:

  1. Work out the area of the two circular faces (2 脳 蟺\(r\)虏).
  2. Work out the curved surface area, this is the rectangular face (2蟺\(r\) 脳 \(h\)).
  3. Sum the area of the circles and the rectangle.

The expression for working out the total surface area of a cylinder is
2蟺\(r\)虏 + 2蟺\(rh\).

\(r\) is the radius of the circular cross-section and \(h\) is the height of the cylinder.
If the diameter, \(d\), of the circular cross-section is given, this is halved to find the radius.

Depending on the orientation of the cylinder, the length of the cylinder is its height.

Example

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  1. A series of two images. The first image shows an upright cylinder. The second image shows the net of the same cylinder. It comprises a rectangle, to represent the curved surface, and two circles attached above and below the rectangle. The circles represent the top and bottom face of the cylinder. The rectangle and curved face of the cylinder are coloured blue. The circles and top and bottom face of the cylinder are coloured orange.

    A cylinder is made up of two congruent circles and a rectangle. The circles are the top face and the base face of the cylinder. The circles are directly opposite each other. The rectangle is the curved face around the cylinder.

Question

Use the formula to work out the surface area of the cylinder.

Use the approximation 蟺 = 3郯14. Give your answer to the nearest mm虏.

An image of a horizontal cylinder. The length of the cylinder is labelled as sixteen millimetres. The diameter of the circular face is labelled as fourteen millimetres. Written above, the formula: surface area equals two pi r squared plus two pi r h.

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Calculate the volume of a cylinder

  • The volume of any cylinder is the area of the cross-section multiplied by its height.

  • This is given by the formula \(V\) = 蟺\(r\)虏\(h\) , where \(r\) is the radius of the circular cross-section and \(h\) is the height of the cylinder.

To work out the volume of a cylinder:

  1. Find the area of the cross-section using the formula for the area of a circle, \(A\) = 蟺\(r\)虏.
  2. Multiply by the height of the cylinder.

Examples

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  1. A series of two images. The first image shows a horizontal cylinder being sliced, vertically, by a knife. There is a dashed curve showing the pathway of the intended cut. The second image shows the result of this vertical cut. The cylinder is in two pieces. Each piece is a cylinder with the same cross sectional area. The circular faces of the cylinder are coloured orange and the curved surface is coloured blue.

    Imagine that a solid cylinder is sliced open. The face that is revealed is called the cross-section. The cross-section of a cylinder is a circle. The cross-section is the same all the way through the cylinder.

Question

What is the volume of the cylinder? Use the approximation 蟺 = 3郯14

An image of a horizontal cylinder. The length of the cylinder is labelled as eighty metres. The diameter of the circular face is labelled as thirty metres. Written above: v equals pi r squared h.

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Practise finding the surface area and volume of cylinders

Practise working out the surface area and volume of cylinders with this quiz. You may need a pen and paper to help you with your answers.

Quiz

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Real-life maths

An image of tinned food. Each tin is a cylinder. Some of the lids have been partially removed to show their contents.
Certain foods are commonly sold in cylinder-shaped tins.

Certain foods, such as baked beans, vegetables, fish and meat, are commonly sold in cylinder-shaped tins. Cylinders pack efficiently into boxes for shipping, taking up approximately 90% of the available space.

Their circular cross-section also means that the tins can withstand pressure when stored. The food contained in them has a long shelf life.

To make the tins accurately, manufacturers need to calculate the surface area plus a small amount of extra area for the seams. The volume of the cylinder shape will determine the quantity of food that can go inside a tin.

An image of tinned food. Each tin is a cylinder. Some of the lids have been partially removed to show their contents.
Certain foods are commonly sold in cylinder-shaped tins.
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Game - Divided Islands

Using your maths skills, help to build bridges and bring light back to the islands in this free game from 成人快手 Bitesize.

Play the Divided Islands game!
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