Frustum of a Cone: Concepts, Formulas and Examples

A frustum of a cone is what remains when the top of a cone is sliced off by a plane parallel to its base — the shape of a bucket, a lampshade or a flower pot. It has two circular ends of different sizes: the larger radius R at the bottom and the smaller radius r at the top, separated by the vertical height h of the frustum. It is one of the most practical solids in the syllabus because so many real containers are frustums.

Frustums matter because buckets, drinking cups, flower pots and many machine parts are frustums. Their volume formula — (1/3)πh(R² + Rr + r²) — looks long, but each term has a meaning: it blends the big circle, the small circle and their geometric middle. Their curved surface area is πl(R + r), where l is the slant height of the frustum.

Key Concepts

• Lower radius (R): the radius of the larger circular end.
• Upper radius (r): the radius of the smaller circular end.
• Vertical height (h): the perpendicular distance between the two circular ends.
• Slant height (l): the slant edge of the frustum, found by l = √[h² + (R − r)²].
• Volume is in cubic units; surface areas are in square units.

Formulas

• Volume of a frustum: V = (1/3)πh(R² + Rr + r²). It can be seen as the big cone's volume minus the small cut-off cone's volume.
• Curved surface area: CSA = πl(R + r), with l = √[h² + (R − r)²].
• Total surface area: TSA = πl(R + r) + πR² + πr² — the curved part plus both circular ends.
• These assume the cut is parallel to the base, so both ends are circles.

Worked Example

A bucket is a frustum with lower radius R = 8 cm, upper radius r = 5 cm and height h = 6 cm. Find its volume. Use π = 22 ÷ 7.

1. Write the formula: V = (1/3) × π × h × (R² + Rr + r²).
2. Compute the bracket: R² = 64, Rr = 8 × 5 = 40, r² = 25; sum = 64 + 40 + 25 = 129.
3. Substitute: V = (1/3) × (22 ÷ 7) × 6 × 129.
4. Simplify (1/3) × 6 = 2, so V = 2 × (22 ÷ 7) × 129 = 44 × 129 ÷ 7 = 5676 ÷ 7 ≈ 810.86. The volume is about 810.86 cm³, roughly 0.81 litres.
5. Slant height check: l = √[36 + (8 − 5)²] = √[36 + 9] = √45 ≈ 6.71 cm, so CSA = (22 ÷ 7) × 6.71 × 13 ≈ 274 cm².

Common Misconceptions

• Using (R + r)² instead of R² + Rr + r²: the middle term is Rr, not 2Rr; (R + r)² would give the wrong volume.
• Forgetting the 1/3: the frustum comes from cones, so the one-third factor of cone volumes carries over.
• Using h directly in the CSA: the curved area needs the slant height l = √[h² + (R − r)²], not the vertical height.

Real-World Applications

• Sizing buckets, tubs, flower pots and drinking cups.
• Designing lampshades and loudspeaker cones.
• Manufacturing tapered machine parts, nozzles and hoppers.
• Estimating concrete for tapered pillars and road dividers.

A frustum is a cone with its tip removed: its volume is (1/3)πh(R² + Rr + r²) and its curved area is πl(R + r). Compute the bracket term by term, find the slant height first, and remember the cut must be parallel to the base.