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.
What Is the Volume of a Cone?
A cone is a 3D shape with a circular base that tapers to a single point (the apex). The perpendicular height is the vertical distance from the base to the apex — not the slant height along the side.
The volume of a cone is one-third of the volume of the cylinder that would enclose it. This relationship is the key to remembering the formula: take the cylinder formula (pi r² h) and divide by 3.
A frustum is the solid left when the top of a cone is cut off by a plane parallel to the base. To find the volume of a frustum, calculate the volume of the full cone and subtract the volume of the small cone that was removed.
Key Formulas
Step-by-Step Method
- Identify the radius (r) and the perpendicular height (h) — not the slant height.
- Substitute into V = ⅓ pi r² h.
- For frustums, find the dimensions of both the large and small cones, then subtract.
Worked Example 1 — Foundation Level
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.
Working: V = ⅓ pi r² h V = ⅓ × pi × 6² × 10 V = ⅓ × pi × 36 × 10 V = ⅓ × 360pi V = 120pi
Answer: 120pi cm³
Worked Example 2 — Higher Level
Question: A cone has a volume of 300pi cm³ and a base radius of 10 cm. Find its perpendicular height.
Working: V = ⅓ pi r² h 300pi = ⅓ × pi × 10² × h 300pi = ⅓ × 100pi × h 300 = ⅓ × 100 × h 300 = (100/3) × h h = 300 × 3 ÷ 100 h = 900 ÷ 100 h = 9
Answer: 9 cm
Worked Example 3 — Exam Style
Question: A frustum is formed by removing a small cone from the top of a large cone. The large cone has base radius 9 cm and perpendicular height 12 cm. The small cone has base radius 3 cm and perpendicular height 4 cm. Find the volume of the frustum to the nearest whole number.
Working: Large cone volume = ⅓ × pi × 9² × 12 = ⅓ × pi × 81 × 12 = ⅓ × 972pi = 324pi Small cone volume = ⅓ × pi × 3² × 4 = ⅓ × pi × 9 × 4 = ⅓ × 36pi = 12pi Frustum volume = 324pi − 12pi = 312pi = 980.176...
Answer: 980 cm³
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.
Practice Questions
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.
Practise volume of a cone questions with instant feedback — completely free on GCSEMathsAI.
Related Topics
Summary
- The volume of a cone is V = ⅓ pi r² h — one-third of the equivalent cylinder volume.
- Always use the perpendicular height, not the slant height.
- A frustum is formed by removing a smaller cone from the top; subtract the small cone volume from the large cone volume.
- The formula is given on the exam formula sheet, but writing it in your working still earns marks.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
Further reading from leading academic institutions — free and open-access.