A pyramid or cone holds exactly one third of the prism it fits inside. Spheres have their own formulas. All of them are on the formula page: the skill is choosing the right one and the right length.
Pyramid \(V = \tfrac{1}{3}Ah\); cone \(V = \tfrac{1}{3}\pi r^2 h\) and curved surface \(\pi r l\); sphere \(V = \tfrac{4}{3}\pi r^3\) and surface \(4\pi r^2\). All are in the List of formulas. The cone's curved surface uses the SLANT height \(l\), where \(l^2 = r^2 + h^2\); that relationship is not given.
Fig. 1: a cone of radius 6 cm and vertical height 8 cm (drawn to scale); the slant height is 10 cm by Pythagoras
Section 01
Pyramids: one third.
In the lab, press Pour three times: it takes exactly three full pyramids (or cones) to fill the prism (or cylinder) with the same base and height. That is where the \(\tfrac{1}{3}\) comes from.
Solid
Volume
Surface area
Pyramid
\(\tfrac{1}{3}Ah\)
base + triangular faces
Cone
\(\tfrac{1}{3}\pi r^2 h\)
curved \(\pi rl\) (+ base \(\pi r^2\))
Sphere
\(\tfrac{4}{3}\pi r^3\)
\(4\pi r^2\)
Worked example Square-based pyramid
A pyramid has a square base of side 9 cm and a vertical height of 10 cm. Find its volume.
Base area \(A = 9 \times 9 = 81\text{ cm}^2\).
\(V = \tfrac{1}{3} \times 81 \times 10 = 270\).
Answer: \(270\text{ cm}^3\)
Worked example Surface area of a pyramid
A square-based pyramid has base edges of 6 cm. Each triangular face has a perpendicular height of 8 cm (measured up the face). Find its total surface area.
Base \(= 6 \times 6 = 36\).
One triangle \(= \tfrac{1}{2} \times 6 \times 8 = 24\); there are four, so \(96\).
Total \(= 36 + 96 = 132\).
Answer: \(132\text{ cm}^2\)
Examiner trap
The \(h\) in \(\tfrac{1}{3}Ah\) is the VERTICAL height from the base to the apex, not the sloping height of a face. Forgetting the \(\tfrac{1}{3}\) gives the volume of the prism instead.
Section 02
Cones and the slant height.
A cone has two different heights (Fig. 1). The vertical height \(h\) goes in the volume; the slant height \(l\) goes in the curved surface area. If you are given one, use Pythagoras to find the other.
Worked example Volume and curved surface
For the cone in Fig. 1 (\(r = 6\) cm, \(h = 8\) cm), find (a) the volume (b) the curved surface area.
Putting the vertical height into \(\pi rl\): for the cone above, \(\pi \times 5 \times 12\) is wrong. The curved surface needs the slant height 13.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Which of these is NOT printed in the 0580 List of formulas?
Quick check+10 XP
A pyramid has base area \(30\text{ cm}^2\) and vertical height 10 cm. What is its volume?
Quick check+10 XP
Find the surface area of a sphere of radius 3 cm, in terms of \(\pi\).
Quick check+10 XP
A cone has radius 3 cm and vertical height 4 cm. What is its slant height?
Section 03
Spheres and hemispheres.
In the lab, change the radius of the sphere: doubling the radius makes the surface area 4 times bigger and the volume 8 times bigger. A hemisphere is half a sphere, plus a flat circular face.
Worked example Sphere
A sphere has radius 6 cm. Find its volume and surface area.