Volume and Surface Area of Prisms
A prism is a 3D shape with a constant cross-section running along its length, such as a cuboid, with a rectangular cross-section, a triangular prism, or a cylinder, with a circular cross-section, and its volume is found using volume = cross-sectional area x length. The surface area of a prism is the total area of every face, found by identifying each face, calculating its area, and adding the areas together.
Before you start
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Method
- Identify the cross-section of the prism: the 2D shape that stays the same all along its length, such as a rectangle, a triangle or a circle.
- Find the area of this cross-section using the correct 2D area formula.
- Find the volume by multiplying the cross-sectional area by the length (or height) of the prism.
- For surface area, identify every face of the solid, including both end faces (the cross-section, counted twice), and find the area of each face separately.
- For a cuboid, remember there are three pairs of equal, opposite rectangular faces.
- Add all the face areas together for the total surface area, and give your final answer with the correct unit: cm^3 for volume, cm^2 for surface area.
Worked example
A triangular prism has a cross-section that is a right-angled triangle with base 6 cm and height 8 cm (hypotenuse 10 cm), and the prism is 15 cm long. Find (a) its volume, (b) its total surface area.
- Area of the triangular cross-section = 1/2 x base x height = 1/2 x 6 x 8 = 24 cm^2.
- Volume = cross-sectional area x length = 24 x 15 = 360 cm^3.
- The two triangular end faces contribute 2 x 24 = 48 cm^2.
- The three rectangular faces (one for each side of the triangle, each multiplied by the 15 cm length) contribute (6 x 15) + (8 x 15) + (10 x 15) = 90 + 120 + 150 = 360 cm^2.
- Total surface area = 48 + 360 = 408 cm^2.
Practice questions
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Q1A cuboid has length 8 cm, width 5 cm, and height 4 cm. Find its volume.Show answer
Answer: 8 x 5 x 4 = 160 cm^3
Q2Find the volume of a cuboid measuring 10 cm by 6 cm by 3 cm.Show answer
Answer: 10 x 6 x 3 = 180 cm^3
Q3A cylinder has a circular cross-section of area 50 cm^2 and a length of 12 cm. Find its volume.Show answer
Answer: 50 x 12 = 600 cm^3
Q4A triangular prism has a triangular cross-section of area 18 cm^2 and a length of 20 cm. Find its volume.Show answer
Answer: 18 x 20 = 360 cm^3
Q5Find the surface area of a cube with side length 5 cm.Show answer
Answer: 6 faces, each 5 x 5 = 25 cm^2, total = 6 x 25 = 150 cm^2
Q6A cuboid has length 6 cm, width 4 cm, and height 3 cm. Find its total surface area.Show answer
Answer: 2(6x4) + 2(6x3) + 2(4x3) = 48 + 36 + 24 = 108 cm^2
Q7A cylinder has radius 3 cm and length 10 cm. Find its volume, using pi = 3.14.Show answer
Answer: Cross-sectional area = 3.14 x 9 = 28.26 cm^2, volume = 28.26 x 10 = 282.6 cm^3
Q8A prism has a volume of 240 cm^3 and a cross-sectional area of 20 cm^2. Find its length.Show answer
Answer: Length = 240 / 20 = 12 cm
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A cuboid has length 9 cm, width 4 cm, and height 5 cm. Work out its total surface area.
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A cylinder has a radius of 5 cm and a height of 12 cm. (a) Work out the area of its circular cross-section, using pi = 3.14. (b) Work out the volume of the cylinder. (c) The cylinder is full of water. A second, identical, empty cylinder is filled by pouring in exactly half of the water from the first cylinder. Work out the volume of water in the second cylinder.
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Free printable worksheet
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This topic is chapter 12 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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