Volume of Pyramids, Cones, and Spheres
Grade 5The formulas
Volume of pyramid = ⅓ × base area × height
Volume of cone = ⅓πr²h
Volume of sphere = (4/3)πr³
The ⅓ in the pyramid and cone formulas reflects the fact that these shapes taper to a point — they hold exactly one-third of the volume of the corresponding prism or cylinder with the same base and height.
The sphere formula
The sphere formula is given on the GCSE formula sheet, but you need to know when to use it and how to rearrange it. The radius must be cubed — a common error is squaring it.
For a cone, h is the perpendicular height — the vertical distance from base to apex. The slant height l is used for surface area, not volume. Do not mix them up.
Worked example
EXAMPLE
A cone has radius 5 cm and perpendicular height 12 cm. Find its volume. Give your answer in terms of π.
Volume = ⅓ × π × 5² × 12 = ⅓ × π × 25 × 12 = ⅓ × 300π = 100π cm³.
Try this one
A solid sphere has a radius of \( 3 \) cm.
Calculate the volume of the sphere. Give your answer to \( 1 \) decimal place. [3]
Nova's hint:
Try to Solve it with Nova?
Use the sphere volume formula, which involves the cube of the radius.
Ask yourself: 'have I cubed the radius rather than squaring it?' Work out \( r^3 \) first, then apply the \( \dfrac{4}{3}\pi \) factor.
