Last updated: 2026-05-20

The cone frustum

A cone frustum — also called a truncated cone — is the solid that remains when the top of a right circular cone is removed by a cut parallel to the base. It has two circular faces (a larger bottom with radius R and a smaller top with radius r), a curved lateral surface connecting them, and a height h measured perpendicularly between the two bases.

Formulas

All four output quantities follow directly from R, r, and h.

Slant height is the straight-line distance along the lateral surface from a point on the bottom edge to the corresponding point on the top edge:

$l = \sqrt{(R - r)^2 + h^2}$

This is just the Pythagorean theorem applied to the right triangle whose legs are the height difference between the two radii and the vertical height.

Volume is derived by subtracting the removed small cone from the original large cone:

$V = \frac{\pi h}{3}(R^2 + Rr + r^2)$

Lateral surface area (the curved side only) unfolds into a flat annular sector with inner and outer arcs:

$A_{\text{lat}} = \pi(R + r)\,l$

Total surface area adds both circular faces:

$A_{\text{tot}} = A_{\text{lat}} + \pi R^2 + \pi r^2 = \pi(R + r)l + \pi(R^2 + r^2)$

Frustum vs. full cone

A frustum is a cone with its tip removed. When the top radius r = 0, the frustum formulas reduce exactly to the standard cone formulas (V = πR²h/3, lateral area = πRl). When r = R, the solid becomes a cylinder: the slant height equals h, the volume equals πR²h, and the lateral area equals 2πRh. Both limiting cases serve as sanity checks when verifying a calculation.

Worked example

A metal bucket measures 15 cm across the top, 22 cm across the bottom, and stands 28 cm tall. Find its volume and total sheet-metal area.

• R = 11 cm, r = 7.5 cm, h = 28 cm
• $l = \sqrt{(11 - 7.5)^2 + 28^2} = \sqrt{12.25 + 784} = \sqrt{796.25} \approx 28.22$ cm
• V = (π × 28/3) × (121 + 82.5 + 56.25) = (π × 28/3) × 259.75 ≈ 7,616 cm³ ≈ 7.62 litres
• A_lat = π × (11 + 7.5) × 28.22 = π × 18.5 × 28.22 ≈ 1,639 cm²
• A_tot = 1,639 + π × 121 + π × 56.25 ≈ 1,639 + 380.1 + 176.7 ≈ 2,196 cm²

The bucket holds about 7.6 litres and requires roughly 2,196 cm² (≈ 0.22 m²) of sheet metal to fabricate — before accounting for overlap at the seam.

Why the volume formula has three terms

The factor (R² + Rr + r²) comes from the algebraic identity used when subtracting the small cone from the large one. If the large cone has apex-to-base height H and the small cone has height H − h, then:

$V_{\text{frustum}} = \frac{\pi H}{3} R^2 - \frac{\pi (H - h)}{3} r^2$

After eliminating H (using the similar-triangles relation H/R = (H − h)/r) and simplifying, the expression collapses to the compact form above. The three terms — R², Rr, and r² — can be interpreted as the areas of the bottom base, the geometric mean base, and the top base respectively, weighted by h/3.

Real-world applications

• Lampshades — the tapered shade is a frustum; its lateral area gives the amount of fabric needed to cover it.
• Industrial silos and hoppers — frustum-shaped hoppers direct bulk materials; volume determines capacity.
• Architecture — truncated-cone roofs, cooling towers, and decorative columns all use frustum geometry.
• Rocketry — nose-cone sections between cylinders and the tip are often modelled as frustums for drag calculations.
• Cups and buckets — paper cups and pails are thin-walled frustums; lateral area equals the flat-blank material area.