Surface Area Calculator

The surface area of a solid object refers to the total area covered by all its outer surfaces. This calculator helps you find the surface area of common 3D shapes using standard formulas. For a deeper understanding of each shape, you can visit our Volume Calculator and Area Calculator pages. Here, we’ll focus on how to calculate surface area and provide real-life examples to help illustrate each formula.

Surface Area of a Sphere

To calculate the surface area (SA) of a sphere, use the formula:

SA = 4πr²
Where:
r = radius of the sphere

Example:
Xael loves her Lindt chocolate truffles and doesn’t like to share. To discourage others from eating them, she calculates the total surface area she’ll lick. For truffles with a radius of 0.325 inches:

SA = 4 × π × (0.325)² ≈ 1.327 in²

Surface Area of a Cone

A cone’s total surface area includes both its base and its slanted surface (lateral area):

Base SA = πr²
Lateral SA = πr√(r² + h²)
Total SA = πr(r + √(r² + h²))
Where:
r = base radius, h = height

Example:
Athena, fascinated by Southeast Asian rice hats, decides to make her own using fabric from an old wedding dress. For a cone with radius 1 ft and height 0.5 ft:

Lateral SA = π × 1 × √(1² + 0.5²) ≈ 4.272 ft²

Surface Area of a Cube

A cube has six identical square faces. The surface area is:

SA = 6a²
Where:
a = edge length

Example:
Anne custom-orders a black Rubik’s Cube for her brother. With edge length 4 inches:

SA = 6 × 4² = 96 in²

Surface Area of a Cylinder (Closed)

For a closed cylinder (with both top and bottom):

Base SA = 2πr²
Lateral SA = 2πrh
Total SA = 2πr(r + h)
Where:
r = radius, h = height

Example:
Jeremy uses a large cylindrical tank to bathe. For a tank with radius 3.5 ft and height 5.5 ft:

SA = 2π × 3.5 × (3.5 + 5.5) ≈ 197.92 ft²

Surface Area of a Rectangular Tank

The surface area of a rectangular box is:

SA = 2lw + 2lh + 2wh
Where:
l = length, w = width, h = height

Example:
Banana wraps a gift in a 3×4×5 ft box. The wrapping paper needed:

SA = 2(3×4 + 4×5 + 3×5) = 94 ft²

Surface Area of a Capsule

A capsule combines a cylinder with two hemispherical ends:

SA = 4πr² + 2πrh
Where:
r = radius, h = cylinder height (not including hemispheres)

Example:
Horatio is preparing sugar-coated capsules. For r = 0.05 in and h = 0.5 in:

SA = 4π × (0.05)² + 2π × 0.05 × 0.5 ≈ 0.188 in²

Surface Area of a Spherical Cap

A spherical cap is a portion of a sphere. Assuming a solid (not hollow), the surface area includes:

Cap SA = 2πRh
Base SA = πr²
Total SA = 2πRh + πr²
Where:
R = cap radius, r = base radius, h = cap height

Example:
Jennifer cuts a section from a globe. For R = 0.80 ft, h = 0.53 ft:

SA = 2π × 0.80 × 0.53 ≈ 2.664 ft²

Surface Area of a Conical Frustum

A conical frustum has two circular ends and a sloped surface:

End SA = π(R² + r²)
Lateral SA = π(R + r)√((R – r)² + h²)
Total SA = π(R² + r² + (R + r)√((R – r)² + h²))
Where:
R and r = radii of ends, h = height

Example:
Paul designs a volcano-shaped model using a closed frustum. For R = 1 ft, r = 0.3 ft, h = 1.5 ft:

SA ≈ 10.185 ft²

Surface Area of an Ellipsoid

There’s no exact simple formula for ellipsoid surface area, but a common approximation is:

SA ≈ 4π × ((a^1.6b^1.6 + a^1.6c^1.6 + b^1.6c^1.6) / 3)^(1/1.6)
Where:
a, b, c = the three semi-axes

Example:
Coltaine slices vegetables into elliptical shapes. For a = 0.1 in, b = 0.2 in, c = 0.35 in:

SA ≈ 0.562 in²

Surface Area of a Square Pyramid

The surface area of a square pyramid includes its square base and four triangular sides:

Base SA = a²
Lateral SA = 2a√((a/2)² + h²)
Total SA = a² + 2a√((a/2)² + h²)
Where:
a = base edge, h = height

Example:
Vonquayla coats her sugar pyramid for a school project. For a = 3 ft and h = 5 ft:

SA ≈ 40.321 ft²

Common Surface Area Units