CalConverters

Conical Frustum Calculator: Volume & Surface Area

Find a conical frustum's slant height, volume, lateral area, and total surface area from its two radii and height, with exact step-by-step results.

Loading calculator…

The conical frustum's four measures

A conical frustum is a cone with its tip cut off by a plane parallel to the base — two circular ends of different radii, bottom r1r_1 and top r2r_2, joined by a tapering side of height hh. Given r1r_1, r2r_2, and hh, every other measure follows:

s=(r1−r2)2+h2s = \sqrt{(r_1 - r_2)^2 + h^2} V=13πh(r12+r22+r1r2)V = \frac{1}{3}\pi h(r_1^2 + r_2^2 + r_1 r_2) L=π(r1+r2)sA=π[r12+r22+(r1+r2)s]L = \pi(r_1 + r_2)s \qquad A = \pi\left[r_1^2 + r_2^2 + (r_1 + r_2)s\right]

The slant height ss is the hypotenuse of a right triangle: one leg is the height hh, the other is the horizontal offset between the two rims, ∣r1−r2∣|r_1 - r_2| — Pythagoras, exactly as with a plain cone's own slant height.

Tip: The slant uses the DIFFERENCE r1−r2r_1 - r_2, squared — never the sum. And the volume's middle term is r1r2r_1 r_2, not 2r1r22r_1r_2 — check both against (r1+r2)2(r_1+r_2)^2 if a result looks off.

Worked example: r1 = 6, r2 = 3, h = 8

s=(6−3)2+82=9+64=73≈8.5440s = \sqrt{(6-3)^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} \approx 8.5440 V=13π(8)(36+9+18)=13π(8)(63)=168π≈527.7876V = \frac{1}{3}\pi(8)(36 + 9 + 18) = \frac{1}{3}\pi(8)(63) = 168\pi \approx 527.7876 L=π(9)(8.5440)≈241.5760A≈382.9477L = \pi(9)(8.5440) \approx 241.5760 \qquad A \approx 382.9477

Notice that the slant height 73\sqrt{73} is irrational (73 has no perfect-square factor), yet the volume still comes out to an EXACT multiple of π\pi (168π168\pi) — the two are independent, since VV never depends on ss at all.

Two limiting cases: cone and cylinder

Tip: Set r2=0r_2 = 0 for a cone, or r1=r2r_1 = r_2 for a cylinder — both are useful sanity checks, since the frustum formulas must reduce to those simpler solids.
  • r1 = 3, r2 = 0, h = 4 (reduces to a cone): s=(3−0)2+16=25=5s = \sqrt{(3-0)^2+16} = \sqrt{25} = 5, V=13π(4)(9)=12π≈37.6991V = \frac{1}{3}\pi(4)(9) = 12\pi \approx 37.6991 — matching the cone calculator at r = 3, h = 4 exactly.
  • r1 = 5, r2 = 5, h = 6 (reduces to a cylinder): s=0+36=6=hs = \sqrt{0+36} = 6 = h, and V=13π(6)(25+25+25)=150πV = \frac{1}{3}\pi(6)(25+25+25) = 150\pi, the same as πr2h\pi r^2 h — matching the cylinder calculator at r = 5, h = 6.

Common mistakes

  • Using (r1+r2)(r_1+r_2) in the slant height. The correct leg is the DIFFERENCE r1−r2r_1 - r_2 (squared, so order never matters) — using the sum overstates the slant.
  • Using (r1+r2)2(r_1+r_2)^2 in the volume. The correct middle term is r1r2r_1r_2 (once), not 2r1r22r_1r_2 — (r1+r2)2(r_1+r_2)^2 silently doubles it.
  • Misspelling "frustum" as "frustrum." The correct spelling has no second "r" after the "t."

Where this shows up

  • Buckets, plant pots, lampshades, paper cups: every one of these everyday containers is a conical frustum — volume tells you capacity, lateral area tells you how much material wraps around the side.
  • Sanity-checking against simpler solids: since r2→0r_2 \to 0 gives a cone and r1=r2r_1 = r_2 gives a cylinder, you can always verify a frustum calculation by checking it against one of those two familiar shapes.

A conical frustum is a cone truncated parallel to its base — knowing the two radii and the height is always enough to determine the slant height, volume, lateral area, and total surface area, with the cone and the cylinder sitting at its two limiting cases.

Frequently asked questions

What are the formulas for a conical frustum?
With bottom radius r1, top radius r2, and height h: slant height s = √((r1 − r2)² + h²), volume V = (1/3)πh(r1² + r2² + r1r2), lateral (side) area L = π(r1 + r2)s, and total surface area A = π[r1² + r2² + (r1 + r2)s].
Why does the slant height use r1 − r2 and not r1 + r2?
The slant is the hypotenuse of a right triangle whose two legs are the height h and the horizontal offset between the rims, |r1 − r2| — Pythagoras, not addition. Squaring removes the sign, so the order of r1 and r2 never matters.
Why does the volume use r1² + r2² + r1r2 instead of (r1 + r2)²?
(r1 + r2)² expands to r1² + r2² + 2·r1·r2 — twice the cross term the real formula needs. Using the sum squared by mistake silently overstates the volume; the correct middle term is a single r1·r2, not 2·r1·r2.
What is a conical frustum?
It's a cone with its tip sliced off by a plane parallel to the base, leaving two circular ends of different radii (r1 at the bottom, r2 at the top) joined by a tapering side. Buckets, lampshades, and paper cups are everyday frustum shapes.
What happens when the top radius r2 is 0?
The frustum reduces to a plain cone — the top circle shrinks to a point (the cone's apex). Every formula here still works: the slant, volume, and surface areas all match the equivalent cone calculator exactly.
What happens when r1 and r2 are equal?
The frustum becomes a cylinder: the slant height equals the height h exactly (since r1 − r2 = 0), and the volume formula simplifies to πr²h, the familiar cylinder volume.
Is the slant height always an exact number?
Not usually — s = √((r1 − r2)² + h²) is irrational unless the radicand happens to be a perfect square. The volume, by contrast, is often an exact multiple of π even when the slant itself is irrational, since V doesn't depend on s at all.
How is the conical frustum related to the cone and the cylinder?
A conical frustum generalizes both: setting r2 = 0 gives a cone, and setting r1 = r2 gives a cylinder. Every conical-frustum formula here reduces to the matching cone or cylinder formula at those two limits.

Related calculators