A frustum is the solid left when a cone's tip is removed by a horizontal cut. At GCSE Higher you find its volume by calculating the complete large cone's volume and subtracting the small removed cone's volume. Similar triangles supply the missing dimension — the height of the small cone — before you can calculate either.
What is a frustum?
A frustum (sometimes spelled frustrum) looks like a bucket or a lampshade — it has two parallel circular faces of different sizes, connected by a slanted curved surface. It is most naturally thought of as a large cone with its tip removed.
The dimensions you need to describe a frustum are:
- R — the radius of the larger (bottom) circular face.
- r — the radius of the smaller (top) circular face.
- h — the perpendicular height of the frustum (distance between the two parallel faces).
The volume of the frustum is found by subtraction: V_frustum = V_large cone − V_small cone.
How do you find the height of the small removed cone?
The small cone that was removed is similar to the large cone — corresponding lengths are in the same ratio. This ratio is r : R (the ratio of the two radii).
If the large cone has height H, then the small cone has height H × (r/R).
The frustum's height h equals the large cone height minus the small cone height: h = H − H(r/R) = H(1 − r/R) = H(R − r)/R.
Rearranging to find H: H = hR/(R − r).
And the small cone height is: H − h = hR/(R − r) − h = h(R − R + r)/(R − r) = hr/(R − r).
How do you calculate the volume of a frustum step by step?
Worked example: A frustum has bottom radius R = 6 cm, top radius r = 3 cm, and perpendicular height h = 8 cm. Find its volume in terms of π.
Step 1: Find the height of the complete large cone.
H = hR/(R − r) = 8 × 6/(6 − 3) = 48/3 = 16 cm.
Step 2: Find the height of the small removed cone.
Height of small cone = H − h = 16 − 8 = 8 cm.
Check using ratio: small cone height = H × r/R = 16 × 3/6 = 8 cm. ✓
Step 3: Calculate the volume of the large cone.
V_large = (1/3)πR²H = (1/3)π × 6² × 16 = (1/3)π × 36 × 16 = (1/3) × 576π = 192π cm³.
Step 4: Calculate the volume of the small cone.
V_small = (1/3)πr²(H − h) = (1/3)π × 3² × 8 = (1/3)π × 9 × 8 = (1/3) × 72π = 24π cm³.
Step 5: Subtract.
V_frustum = 192π − 24π = 168π cm³.
As a decimal: 168π ≈ 527.8 cm³.
Can you verify the result with the direct formula?
There is a direct formula for the volume of a frustum: V = (πh/3)(R² + Rr + r²).
Using our values: V = (π × 8/3)(6² + 6 × 3 + 3²) = (8π/3)(36 + 18 + 9) = (8π/3)(63) = 504π/3 = 168π cm³. ✓
At GCSE you are NOT expected to memorise this direct formula — the subtraction method using cone volumes is the standard approach. However, the formula is a useful check if you have time.
Summary of the method
| Step | What to calculate |
|---|---|
| 1 | Height of large cone: H = hR/(R − r) |
| 2 | Height of small cone: H − h |
| 3 | V_large = (1/3)πR²H |
| 4 | V_small = (1/3)πr²(H − h) |
| 5 | V_frustum = V_large − V_small |
How do you spot a frustum question in a GCSE paper?
Frustum questions usually show a 3D shape that looks like a bucket or truncated cone. Look for:
- Two different radii (or diameters) labelled.
- A height for the frustum (not for the complete cone).
- A cross-section diagram showing that the shape has a flat top and a flat bottom.
The question may give the height of the complete cone rather than the frustum height — in that case, you find the small cone height by subtraction and proceed as above.
Frequently asked questions
Do I need to memorise the direct frustum formula for GCSE?
No. The exam board specifications at GCSE (AQA, Edexcel, OCR) expect you to use the cone formula V = (1/3)πr²h and the concept of similar shapes to reconstruct and subtract. The direct formula V = (πh/3)(R² + Rr + r²) is not listed on the GCSE formula sheet and you will not be asked to recall it.
What if the radii given are actually diameters?
Halve them both before starting. If a question states "top diameter 6 cm, bottom diameter 12 cm", then r = 3 cm and R = 6 cm. Always convert to radii before substituting into any volume or surface area formula involving circles.
How does the similar triangles step work in more detail?
The small cone and the large cone share the same apex (tip). A cross-section through the apex gives two similar triangles: one for the full cone and one for the small cone. Because they are similar, every pair of corresponding lengths is in the same ratio. The ratio of the radii (r : R) equals the ratio of the heights (height of small cone : H). Writing this as a fraction: height of small cone = H × r/R.
Is the frustum topic on both Foundation and Higher tiers?
At most exam boards, frustum volume is a Higher-tier topic. It requires combining the cone volume formula with similar triangles and multi-step calculation, which places it firmly in Higher content. Foundation candidates are expected to know the volumes of prisms, cylinders, and sometimes cones, but not frustums.
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