Mathwords logoMathwords

Truncated Cone — Definition, Formula & Examples

A truncated cone is the solid that remains when a smaller cone is cut off the top of a larger cone by a plane parallel to the base. It has two circular faces of different radii and a slanted lateral surface.

A truncated cone (or conical frustum) is the portion of a right circular cone contained between two parallel planes that both intersect the cone, producing two circular cross-sections of radii RR and rr (R>rR > r) separated by a perpendicular height hh.

Key Formula

V = \frac{\pi h}{3}\left(R^2 + Rr + r^2\right)$$ $$A_{\text{lateral}} = \pi(R + r)\,s$$ $$s = \sqrt{h^2 + (R - r)^2}
Where:
  • RR = Radius of the larger base
  • rr = Radius of the smaller base
  • hh = Perpendicular height between the two bases
  • ss = Slant height along the lateral surface

Worked Example

Problem: A truncated cone has a larger base radius of 6 cm, a smaller base radius of 3 cm, and a height of 8 cm. Find its volume and lateral surface area.
Find the volume: Substitute R = 6, r = 3, and h = 8 into the volume formula.
V=π(8)3(62+6⋅3+32)=8π3(36+18+9)=8π3(63)=168π≈527.8 cm3V = \frac{\pi(8)}{3}\left(6^2 + 6 \cdot 3 + 3^2\right) = \frac{8\pi}{3}(36 + 18 + 9) = \frac{8\pi}{3}(63) = 168\pi \approx 527.8 \text{ cm}^3
Find the slant height: Use the Pythagorean relationship between h, the difference in radii, and s.
s=82+(6−3)2=64+9=73≈8.544 cms = \sqrt{8^2 + (6 - 3)^2} = \sqrt{64 + 9} = \sqrt{73} \approx 8.544 \text{ cm}
Find the lateral surface area: Apply the lateral area formula with R + r = 9 and the slant height found above.
Alateral=π(6+3)73=9π73≈241.5 cm2A_{\text{lateral}} = \pi(6 + 3)\sqrt{73} = 9\pi\sqrt{73} \approx 241.5 \text{ cm}^2
Answer: The volume is 168π≈527.8168\pi \approx 527.8 cm³ and the lateral surface area is 9π73≈241.59\pi\sqrt{73} \approx 241.5 cm².

Why It Matters

Truncated cones appear in everyday objects like buckets, lampshades, and paper cups. Engineers use the frustum volume formula when designing tapered tanks and calculating material for conical hoppers. It also shows up frequently on AP and SAT-level geometry problems.

Common Mistakes

Mistake: Using the cylinder formula with an averaged radius instead of the frustum formula.
Correction: The volume formula includes the cross-term RrRr because the radius changes linearly along the height. A simple average-radius cylinder gives the wrong volume; always use V=πh3(R2+Rr+r2)V = \frac{\pi h}{3}(R^2 + Rr + r^2).