Surface area and volume of prisms

Part ofMathsPerimeter, Area, Volume

Key points

A series of four images. Each image shows a triangular prism with a right angled triangle as the cross section. The cross section runs from the front face to the back face of the shape with the right angle in the lower left vertex. The first image has the left face coloured grey, labelled 1. The second image has the front face coloured grey, labelled 2. The third image has the slanted face coloured grey, labelled 3. The fourth image has the base face coloured grey, labelled 4.
A prism has a constant cross-section throughout its length.

  • A has a constant . The cross-section is a .

  • The is made up of at either end of the prism and a set of rectangles between them. The number of rectangular faces is the same as the number of of the shape at each end of the prism.

  • Understanding of shapes and the area of different shapes helps when working out the surface area of a prism. Surface area is measured in square units, such as cm² and mm².

  • The of a prism is the area of its cross-section multiplied by the length. Volume is measured in cubed units, such as cm³ and mm³.

A series of four images. Each image shows a triangular prism with a right angled triangle as the cross section. The cross section runs from the front face to the back face of the shape with the right angle in the lower left vertex. The first image has the left face coloured grey, labelled 1. The second image has the front face coloured grey, labelled 2. The third image has the slanted face coloured grey, labelled 3. The fourth image has the base face coloured grey, labelled 4.
A prism has a constant cross-section throughout its length.
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How to calculate the surface area of a prism

  • A can be named by the shape of its .

    • When the cross-section is a triangle, the prism is called a triangular prism.
    • When the cross-section is a hexagon, the prism is called a hexagonal prism.

  • A is not a prism. The cross-section of a prism is a polygon, a shape bounded by straight lines. A circle is not a polygon.

The surface area is made up of the end faces and rectangular faces that join them.

  • To calculate the total surface area of a prism:
  1. Find the area of the two end faces.
  2. Work out the area of all the rectangular faces in one of two ways:
    • Work out the area of each rectangle separately, length × width.
    • Multiply the perimeter of the end face by the length of the prism.
  3. Sum the areas of all the faces.

Examples

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  1. A series of four images. Each image shows a three dimensional shape. The first and second images are triangular prisms. The first image has a right angled triangle for its cross section. The second image has an equilateral triangle for its cross section. Written below: triangular prisms. The third image is a pentagonal prism and has a pentagon for its cross section. Written below: pentagonal prism. The fourth image is a cylinder and has a circle for its cross section. Written below: cylinder. There is a green tick next to the triangular and pentagonal prisms. The triangular and pentagonal cross sections are coloured green. There is a red cross next to the cylinder. The circle is coloured pink.

    A prism has a uniform cross-section. That means that the cross-section is the same throughout the length of the prism. The polygon shape of the cross-section may be used to name the prism. A prism with a triangle-shaped cross-section is a triangular prism. A prism with a pentagon-shaped cross-section is a pentagonal prism. A 3D shape with a circular cross-section is a cylinder. A cylinder is not a prism because the circle is not a polygon.

Questions

Question 1: Find the total surface area of the prism.

An image of a prism. The cross section of the prism is an isosceles trapezium. The two parallel sides of the trapezium are labelled with dimensions eight millimetres and two millimetres. The two sloping edges have length five millimetres. The area of the cross section has been labelled as twenty millimetres squared. The length of the prism is labelled as ten millimetres. The cross section is coloured grey.

Question 2: Find the total surface area of the triangular prism.

An image of a triangular prism with a right angled triangle as the cross section. The cross section runs from the front face to the back face of the shape with the right angle in the lower left vertex. The dimensions of the triangle have been labelled. It has five of length three centimetres, base of length twelve centimetres, and a diagonal side of length thirteen centimetres. The length of the prism is labelled as ten centimetres. The cross section is coloured grey.

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How to calculate the volume of a prism

The formula for the of a prism is:

\(Volume =\) \(Area\) \(of\) \(cross\)-\(section\) × \(length\)

To calculate the volume of a prism:

  1. Work out the area of the .
  2. Multiply by the length (or height) of the prism.

Examples

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  1. A series of four images. Each image shows a prism. The first image is a triangular prism and has an equilateral triangle for its cross section. The length of the prism has been marked with an arrow and labelled, length. The second image is a pentagonal prism and has a pentagon for its cross section. The length of the prism has been marked with an arrow and labelled, length. The third image is a hexagonal prism and has a hexagon for its cross section. The length of the prism has been marked with an arrow and labelled, length. The fourth image is an upright, hexagonal prism and has a hexagon for its cross section. The height of the prism has been marked with an arrow and labelled, length or height. Written above, the formula: volume equals area of cross section multiplied by length. Written below, a key: orange equals the area of the cross section. Each cross section of the prism is coloured orange. The words, length, length or height are coloured blue. In the formula the words, area of the cross section are coloured orange.

    The volume of a prism is the area of the cross-section multiplied by the length. The length may be the height, depending on the orientation of the prism.

Question

Find the volume of the prism.

An image of a pentagonal prism. The area of the cross section has been labelled as twenty centimetres squared. The length of the prism is labelled as fifteen centimetres. The cross section is coloured grey.

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Practise finding the surface area and volume of prisms

Practise finding the surface area and volume of prisms 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 a pair of spectacles and case.
Manufacturers often use prism shapes for containers or packaging.

Manufacturers often use prism-shaped containers for their products. Triangular prisms and hexagonal prisms are popular choices for packaging for chocolate or cakes, for example, or for gift boxes and glasses cases.

In order to create the prism-shaped boxes, the surface area is designed with a little extra added on. This allows for tabs that are glued or fixed to hold the box or container together when it is folded up, making the complete prism shape.

An image of a pair of spectacles and case.
Manufacturers often use prism shapes for containers or packaging.
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