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Conical Frustum Surface Area Calculator

Calculate the total and lateral surface area, slant height, and volume of a conical frustum from its radii and height.

Top radius (r)

Bottom radius (R)

Height (h)

Enter the top radius, bottom radius, and height — surface areas, slant height, and volume update instantly.

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Everyday Uses

Buckets and planters

Most buckets and flower pots are frustums — find how much soil, water, or paint they really hold.

Lampshade projects

Making or recovering a lampshade? The lateral surface area tells you exactly how much fabric to cut.

Cups and packaging

Paper cups are frustums — calculate their true capacity or the material needed to make them.

Construction transitions

Size funnel sections, chimney caps, and duct reducers that transition between two diameters.

Hoppers and spoil heaps

A truncated cone is the standard hopper and silo outlet shape, and loose material poured on the ground forms one too. Volume gives the batch it holds or the load to shift.

Cutting the flat pattern

A frustum unrolls into a curved annular strip, not a rectangle. Getting that pattern right is the entire difficulty in making one from sheet metal or card.

Frequently Asked Questions

How do you calculate the surface area of a conical frustum?

First find the slant height s = √((R − r)² + h²), where R is the bottom radius, r is the top radius, and h is the vertical height. The lateral (side) surface is π(R + r)s, and the total surface area adds the two circular ends: SA = π(R² + r² + (R + r)s). Example: with r = 3, R = 6, h = 8, the slant height is √(9 + 64) ≈ 8.544, so SA = π(36 + 9 + 9 × 8.544) ≈ 383.0 square units.

What is a conical frustum?

A conical frustum is what remains when the top of a cone is sliced off parallel to its base — leaving a shape with two circular ends of different sizes. Everyday examples include buckets, lampshades, drinking cups, funnels, and volcano-shaped landforms. Setting the top radius to zero turns the frustum back into a full cone.

What is the difference between slant height and vertical height?

Vertical height (h) is the straight up-and-down distance between the two circular ends. Slant height (s) is measured along the sloped side surface, and is always longer whenever the radii differ: s = √((R − r)² + h²). Surface-area formulas use the slant height; volume formulas use the vertical height. If R = r the shape is a cylinder and s = h.

How do you find the volume of a conical frustum?

The volume is V = (πh/3)(R² + Rr + r²). Example: with r = 3, R = 6, and h = 8, V = (8π/3)(36 + 18 + 9) = (8π/3)(63) ≈ 527.8 cubic units. This formula is the difference between the volumes of the full cone and the smaller cone that was cut off the top.