Sheet № 104 · Higher only · AQA · Edexcel · OCR
Volume of a Cone –
The volume of a cone is a Higher-tier topic that builds directly on the volume of a cylinder. A cone holds exactly one-third the volume of a cylinder with the same base radius and height. This guide covers the formula, reverse problems, and the frustum — a favourite of GCSE exam boards including AQA, Edexcel and OCR.
§Key definitions
Question:
A cone has a base radius of 6 cm and a perpendicular height of 10 cm. Find its volume. Give your answer in terms of pi.
Answer:
120pi cm³
Q1 (Foundation):
A cone has radius 5 cm and height 12 cm. Find its volume to 1 decimal place.
Q2 (Foundation):
A cone has radius 3 cm and perpendicular height 7 cm. Find its volume in terms of pi.
Q3 (Higher):
A frustum has a large cone with radius 8 cm and height 15 cm, and the removed small cone has radius 4 cm and height 7.5 cm. Find the volume of the frustum to 3 significant figures.
§Formulas to memorise
V = ⅓ pi r² h, where r is the base radius and h is the perpendicular height
Volume of a frustum = volume of large cone minus volume of small cone
Substitute into V = ⅓ pi r² h.
V = ⅓ pi r² h
V = ⅓ × pi × 6² × 10
V = ⅓ × pi × 36 × 10
V = ⅓ × 360pi
V = 120pi
h = 300 × 3 ÷ 100
h = 900 ÷ 100
Worked example
A cone has a base radius of 6 cm and a perpendicular height of 10 cm. Find its volume. Give your answer in terms of pi.
Working: V = ⅓ pi r² h V = ⅓ × pi × 6² × 10 V = ⅓ × pi × 36 × 10 V = ⅓ × 360pi V = 120pi
⚠ Common mistakes
- ✗Forgetting the one-third. This is the most frequent error. Without the ⅓, you calculate the volume of a cylinder instead of a cone.
- ✗Using the slant height instead of the perpendicular height. The formula requires the vertical height h, not the slant height l. If only the slant height is given, use Pythagoras: h² = l² − r².
- ✗Frustum errors — subtracting the wrong cone. Always subtract the smaller cone (the removed piece) from the larger cone. Check your dimensions carefully using similar triangles.
✦ Exam tips
- →The cone volume formula is on the formula sheet. Write it out at the start of your working to earn the method mark.
- →For frustum questions, set up both cone volumes separately before subtracting. This keeps your working clear and avoids sign errors.
- →If you are given the slant height, draw a right-angled triangle with the radius, perpendicular height, and slant height and use Pythagoras to find the missing measurement.