You are viewing a free preview of this lesson.
Subscribe to unlock all 10 lessons in this course and every other course on LearningBro.
This lesson covers the volume and surface area of 3D shapes for AQA GCSE Mathematics. You need to know the formulae for cuboids, prisms, cylinders, pyramids, cones and spheres. Higher-tier students must also handle frustums and compound solids. These topics are heavily tested and carry significant marks on the exam.
Volume is the amount of 3D space a solid occupies. It is measured in cubic units (e.g. cm cubed, m cubed).
Volume = length x width x height
A cuboid has dimensions 5 cm by 4 cm by 3 cm. Find its volume.
Solution: Volume = 5 x 4 x 3 = 60 cm cubed
A prism is any 3D shape with a uniform cross-section.
Volume of a prism = area of cross-section x length
A triangular prism has a cross-section that is a right-angled triangle with base 6 cm and height 4 cm. The prism is 10 cm long. Find its volume.
Solution: Area of cross-section = (1/2) x 6 x 4 = 12 cm squared Volume = 12 x 10 = 120 cm cubed
Exam Tip: A cuboid is a special type of prism where the cross-section is a rectangle. A cylinder is a prism where the cross-section is a circle. Always think "area of the front face times the depth."
Volume = pi x r squared x h
A cylinder has radius 5 cm and height 12 cm. Find its volume to 1 decimal place.
Solution: Volume = pi x 5 squared x 12 = pi x 25 x 12 = 300pi = 942.5 cm cubed (1 d.p.)
Volume = (1/3) x base area x vertical height
A square-based pyramid has a base of side 6 cm and a vertical height of 10 cm. Find its volume.
Solution: Base area = 6 x 6 = 36 cm squared Volume = (1/3) x 36 x 10 = 360 / 3 = 120 cm cubed
Volume = (1/3) x pi x r squared x h
Note: A cone is essentially a "circular pyramid."
A cone has radius 4 cm and height 9 cm. Find its volume to 1 decimal place.
Solution: Volume = (1/3) x pi x 4 squared x 9 = (1/3) x pi x 16 x 9 = (1/3) x 144pi = 48pi = 150.8 cm cubed (1 d.p.)
Exam Tip: The formula for a cone is given on the formula sheet, but you should still memorise it. Notice that the volume of a cone is exactly one-third the volume of a cylinder with the same radius and height.
Volume = (4/3) x pi x r cubed
A sphere has radius 6 cm. Find its volume to the nearest whole number.
Solution: Volume = (4/3) x pi x 6 cubed = (4/3) x pi x 216 = 288pi = 905 cm cubed (nearest whole number)
| Shape | Formula |
|---|---|
| Cuboid | l x w x h |
| Prism | area of cross-section x length |
| Cylinder | pi x r squared x h |
| Pyramid | (1/3) x base area x h |
| Cone | (1/3) x pi x r squared x h |
| Sphere | (4/3) x pi x r cubed |
A frustum is the shape that remains when the top of a cone or pyramid is cut off with a horizontal slice.
Volume of a frustum = Volume of the large cone - Volume of the small cone removed
A cone has radius 9 cm and height 12 cm. A smaller cone with radius 3 cm and height 4 cm is removed from the top. Find the volume of the frustum.
Solution: Volume of large cone = (1/3) x pi x 9 squared x 12 = (1/3) x pi x 81 x 12 = 324pi
Volume of small cone = (1/3) x pi x 3 squared x 4 = (1/3) x pi x 9 x 4 = 12pi
Volume of frustum = 324pi - 12pi = 312pi = 980.2 cm cubed (1 d.p.)
Exam Tip: Frustum questions are Higher only and often carry 4-5 marks. You need to work out the dimensions of the removed cone using similar triangles if they are not given directly.
Surface area is the total area of all the faces (or surfaces) of a 3D shape.
SA = 2(lw + lh + wh)
A cuboid measures 8 cm by 5 cm by 3 cm. Find its surface area.
Solution: SA = 2(8x5 + 8x3 + 5x3) = 2(40 + 24 + 15) = 2(79) = 158 cm squared
SA = 2 x pi x r squared + 2 x pi x r x h = 2 x pi x r (r + h)
This consists of: 2 circular ends + 1 curved surface (which unrolls into a rectangle).
A cylinder has radius 4 cm and height 10 cm. Find its surface area to 1 decimal place.
Solution: SA = 2 x pi x 4(4 + 10) = 2 x pi x 4 x 14 = 112pi = 351.9 cm squared (1 d.p.)
SA = pi x r x l + pi x r squared
where l is the slant height (not the vertical height).
If you are given the vertical height h and the radius r, find the slant height using Pythagoras: l = square root of (r squared + h squared).
A cone has radius 5 cm and slant height 13 cm. Find its total surface area.
Solution: Curved surface area = pi x 5 x 13 = 65pi Base area = pi x 5 squared = 25pi Total SA = 65pi + 25pi = 90pi = 282.7 cm squared (1 d.p.)
SA = 4 x pi x r squared
A sphere has diameter 14 cm. Find its surface area.
Subscribe to continue reading
Get full access to this lesson and all 10 lessons in this course.