Surface Area Formulas for Common 3D Shapes

This guide walks through the surface area formulas for the ten most-used 3D shapes — what each variable means, how the formula is put together, and a worked example you can sanity-check against the calculator.

Sphere

A sphere is the simplest closed surface. Its area is SA = 4πr², where r is the radius. Tip: a sphere's surface is exactly four times the area of its equatorial cross-section (πr²).

Example. A football with a 0.11 m radius has SA = 4π × 0.11² ≈ 0.152 m².

Cube

A cube has 6 identical square faces, so SA = 6a². For a 4-inch cube, SA = 6 × 16 = 96 in².

Cone (closed)

A closed cone has a circular base and a curved 'lateral' surface that wraps from the base to the apex. The slant height is s = √(r² + h²). The total surface area is SA = πr² + πr·s, i.e. base + lateral.

Cylinder (closed)

A closed cylinder has two circular ends and a rectangular side wall (when 'unrolled'). SA = 2πr² + 2πrh.

Rectangular Tank (cuboid)

A rectangular tank has 6 rectangular faces in three matching pairs. SA = 2(lw + lh + wh).

Capsule

A capsule is a cylinder with two hemispherical ends. The two hemispheres together make a full sphere (4πr²) and the side wall adds 2πrh, giving SA = 4πr² + 2πrh.

Spherical Cap

A spherical cap is a 'dome' — the part of a sphere on one side of a flat plane. The cap is defined by any two of: base radius r, ball radius R, cap height h, with the relation r² = h(2R − h). The total surface area (curved cap plus flat base disc) is SA = 2πRh + πr².

Conical Frustum

A conical frustum is a cone with the top cut off — like a bucket or a paper cup. With top radius r₁, bottom radius r₂ and height h, the slant length is s = √((r₂−r₁)² + h²). Total surface area is SA = π(r₁ + r₂)·s + πr₁² + πr₂².

Ellipsoid (approximation)

An ellipsoid has three semi-axes a, b, c. There is no closed-form expression for its surface area, but the Knud Thomsen approximation is excellent: SA ≈ 4π·((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p) with p = 1.6075. Maximum error is around 1.06%.

Square Pyramid (closed)

A square pyramid has one square base and 4 triangular faces. With base edge a and height h, the slant height of each triangle is s = √((a/2)² + h²), so each triangle has area (a·s)/2. Total SA = a² + 2a·s.

Mixing units

You can mix units within a calculation — e.g. radius in inches and height in feet — by converting everything to a common unit first. The calculator does this automatically by converting each input to metres before applying the formula.

Open vs closed shapes

Most of the formulas above assume a closed solid. If you only need the side wall of a cylinder (an open tube), use 2πrh. If you need an open-top tank or pyramid, simply subtract the missing face's area.

Try the calculator

Use our Surface Area Calculator to apply any of these formulas in any unit, see the breakdown, and switch units freely on input and output.

Sources

• Standard geometric formulas (Stewart, "Calculus"; Bronshtein & Semendyayev, "Handbook of Mathematics").
• Knud Thomsen — ellipsoid surface-area approximation, p = 1.6075.
• NIST — SI and Imperial unit conversions.

Last reviewed: April 2026.