Surface Area Calculator

What is Surface Area?

Surface area is the total area that the surface of a three-dimensional object occupies. It is a fundamental measurement in geometry, engineering, architecture, and many scientific fields. Calculating surface area helps determine how much material is needed to cover an object, how heat transfers across its surface, or how it interacts with its environment. For capacity calculations, try our Volume Calculator and Area Calculator.

Our surface area calculator supports ten common 3D shapes, including spheres, cones, cubes, cylinders, rectangular tanks, capsules, spherical caps, conical frustums, ellipsoids, and square pyramids. Each shape has its own formula based on its geometric properties.

Surface Area Formulas

Sphere

A sphere is a perfectly round three-dimensional object. Its surface area is calculated as: $$SA = 4\pi r^2$$ where r is the radius of the sphere. The sphere has the smallest surface area for a given volume of any solid shape, which is why soap bubbles form spheres.

Cone

A cone has a circular base and tapers to a point (apex). Its total surface area includes both the base and the lateral surface: $$SA = \pi r(r + \sqrt{r^2 + h^2})$$ where r is the base radius and h is the height. The term \(\sqrt{r^2 + h^2}\) represents the slant height of the cone.

Cube

A cube has six identical square faces. Its surface area is simply: $$SA = 6a^2$$ where a is the length of any edge. This is one of the simplest surface area calculations.

Cylinder

A cylinder has two circular bases and a curved lateral surface: $$SA = 2\pi r(r + h)$$ where r is the base radius and h is the height. This formula combines the areas of the two circular ends (\(2\pi r^2\)) and the lateral surface (\(2\pi rh\)).

Rectangular Tank

A rectangular tank (or rectangular prism) has six rectangular faces: $$SA = 2lw + 2lh + 2wh$$ where l is the length, w is the width, and h is the height.

Capsule

A capsule consists of a cylinder with hemispherical ends: $$SA = 4\pi r^2 + 2\pi rh$$ where r is the radius and h is the height of the cylindrical portion. The \(4\pi r^2\) term accounts for the two hemispheres (which together form a full sphere).

Spherical Cap

A spherical cap is the portion of a sphere cut off by a plane: $$SA = 2\pi Rh + \pi r^2$$ where R is the radius of the sphere, r is the radius of the base of the cap, and h is the height of the cap.

Conical Frustum

A conical frustum is the portion of a cone between two parallel cutting planes: $$SA = \pi(R^2 + r^2) + \pi(R + r)\sqrt{(R - r)^2 + h^2}$$ where R is the bottom radius, r is the top radius, and h is the height.

Ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it using directional scaling. Its surface area is approximated by: $$SA \approx 4\pi\left(\frac{a^{1.6}b^{1.6} + a^{1.6}c^{1.6} + b^{1.6}c^{1.6}}{3}\right)^{\frac{1}{1.6}}$$ where a, b, and c are the semi-axes lengths. This is an approximation formula for near-spherical ellipsoids.

Square Pyramid

A square pyramid has a square base and four triangular faces meeting at an apex: $$SA = a^2 + 2a\sqrt{(a/2)^2 + h^2}$$ where a is the base edge length and h is the height.