Surface Area Calculator

The surface area of a solid is a measure of the total area occupied by the surface of an object. All of the objects addressed in this calculator are described in more detail on the Volume Calculator and Area Calculator pages. As such, this calculator will focus on the equations for calculating the surface area of the objects and the use of these equations. Please refer to the aforementioned calculators for more detail on each individual object.

Sphere

The surface area (SA) of a sphere can be calculated using the equation:

SA = 4πr²
where r is the radius

For example, a spherical chocolate truffle with a radius of 0.325 inches has a surface area of:

SA = 4 × π × 0.325² = 1.327 in²

Cone

The surface area of a circular cone can be calculated by summing the surface area of each of its individual components. The "base SA" refers to the circle that comprises the base of a closed circular cone, while the lateral SA refers to the rest of the area of the cone between the base and its apex. The equations to calculate each, as well as the total SA of a closed circular cone, are:

base SA = πr²
lateral SA = πr√r² + h²
total SA = πr(r + √r² + h²)
where r is the radius and h is the height

For example, a conical hat with a radius of 0.4 feet and a height of 0.5 feet has a lateral surface area of:

lateral SA = π × 0.4√0.4² + 0.5² = 0.805 ft²

Cube

The surface area of a cube can be calculated by summing the total areas of its six square faces:

SA = 6a²
where a is the edge length

For example, a cube with an edge length of 4 inches has a surface area of SA = 6 × 4² = 96 in².

Cylindrical Tank

The surface area of a closed cylinder can be calculated by summing the total areas of its base and lateral surface:

base SA = 2πr²
lateral SA = 2πrh
total SA = 2πr(r + h)
where r is the radius and h is the height

For example, a cylindrical tank with a radius of 3.5 feet and a height of 5.5 feet has a total surface area of 2π × 3.5(3.5 + 5.5) = 197.920 ft².

Rectangular Tank

The surface area of a rectangular tank is the sum of the area of each of its faces:

SA = 2lw + 2lh + 2wh
where l is the length, w is the width, and h is the height

For example, wrapping a box of 3 ft × 4 ft × 5 ft takes SA = (2 × 3 × 4) + (2 × 4 × 5) + (2 × 3 × 5) = 94 ft² of paper.

Capsule

The surface area of a capsule can be determined by combining the surface area equations for a sphere and the lateral surface area of a cylinder. Note that the surface area of the bases of the cylinder is not included since it does not comprise part of the surface area of a capsule. The total surface area is calculated as follows:

SA = 4πr² + 2πrh
where r is the radius and h is the height

For example, a capsule with r of 0.05 inches and h of 0.5 inches has SA = 4π × 0.05² + 2π × 0.05 × 0.5 = 0.188 in².

Spherical Cap

The surface area of a spherical cap is based on the height of the segment in question. The calculator provided assumes a solid sphere and includes the base of the cap in the calculation of surface area, where the total surface area is the sum of the area of the base and that of the lateral surface of the spherical cap. If using this calculator to compute the surface area of a hollow sphere, subtract the surface area of the base. Given two values of height, cap radius, or base radius, the third value can be calculated using the equations provided on the Volume Calculator. The surface area equations are as follows:

spherical cap SA = 2πRh
base SA = πr²
Total solid sphere SA = 2πRh + πr²
where R is the spherical cap radius, r is the base radius, and h is the height

For example, a hollow portion of a globe with R of 0.80 feet and h of 0.53 feet has a cap surface area of 2π × 0.80 × 0.53 = 2.664 ft².

Conical Frustum

The surface area of a solid, right conical frustum is the sum of the areas of its two circular ends and that of its lateral face:

circular end SA = π(R² + r²)
lateral SA = π(R + r)√(R - r)² + h²
total SA = π(R² + r²) + π(R + r)√(R - r)² + h²
where R and r are the radii of the ends, and h is the height

For example, a closed conical frustum with R of 1 foot, r of 0.3 feet, and h of 1.5 feet has a total surface area of 10.185 ft².

Ellipsoid

Calculating the surface area of an ellipsoid does not have a simple, exact formula such as a cube or other simpler shape does. The calculator above uses an approximate formula (the Knud Thomsen approximation) that assumes a nearly spherical ellipsoid:

SA ≈ 4π ^1.6√(a^1.6b^1.6 + a^1.6c^1.6 + b^1.6c^1.6)/3
where a, b, and c are the axes of the ellipse

For example, an elliptical cut with axes 0.1, 0.2, and 0.35 inches has a surface area of approximately 0.562 in².

Square Pyramid

The surface area of a square pyramid is comprised of the area of its square base and the area of each of its four triangular faces. Given height h and edge length a, the surface area can be calculated using the following equations:

base SA = a²
lateral SA = 2a√(a/2)² + h²
total SA = a² + 2a√(a/2)² + h²

For example, a model pyramid with edge length a of 3 feet and height h of 5 feet has a total surface area of 3² + 2 × 3√(3/2)² + 5² = 40.321 ft².

Common Area Units