Surface Area Calculator (Multi-Solid)

This calculator works out the total and lateral surface area of seven common solids: a sphere, cube, rectangular box (cuboid), cylinder, cone, square pyramid or triangular prism. Surface area matters whenever you need to know how much material covers an object, from paint or wrapping paper to sheet metal or fabric, and it also feeds into packaging design and heat-loss calculations. Choose the solid shape and your unit of measurement (centimetres, metres, millimetres, inches or feet), then enter the dimensions that shape needs, for example the radius of a sphere, the side length of a cube, or the base, triangle height, both slant sides and length of a triangular prism. As you type, the calculator instantly returns the total surface area (highlighted) and the lateral surface area, which counts only the curved or side faces and excludes the top and bottom. It also shows the exact formula used for your chosen shape, plus a breakdown of how each face contributes to the total, such as the two circular bases and curved wall of a cylinder, or the square base and four triangular faces of a pyramid. Slant heights for cones and pyramids are worked out automatically using the Pythagorean theorem. Results update live to two decimal places in your chosen unit squared. These figures come from standard geometric formulae for regular solids, so treat irregular or composite shapes as needing separate calculations for each component.

Surface Area Result

Shape

Sphere

Selected solid

Total Surface Area
314.16
cm²

Lateral Surface Area

314.16

cm2 (curved/side surface)

Surface Area Formulae for Common Solids

Surface area is the total area of all faces (or the curved surface) of a three-dimensional object. It is measured in square units. Below are the standard formulae used by this calculator.

Shape	Total Surface Area Formula	Key Variables
Sphere	4 x pi x r²	r = radius
Cube	6 x a²	a = side length
Rectangular Box (Cuboid)	2(lw + lh + wh)	l = length, w = width, h = height
Cylinder	2 x pi x r x (r + h)	r = radius, h = height
Cone	pi x r x (r + l), where l = sqrt(r² + h²)	r = base radius, h = height, l = slant height
Square Pyramid	a² + 2 x a x l, where l = sqrt((a/2)² + h²)	a = base side, h = height, l = slant height
Triangular Prism	(b x h_t) + L x (b + s1 + s2)	b = triangle base, h_t = triangle height, L = prism length, s1/s2 = triangle sides

Shape

Total Surface Area Formula

Key Variables

Sphere

4 x pi x r²

r = radius

Cube

6 x a²

a = side length

Rectangular Box (Cuboid)

2(lw + lh + wh)

l = length, w = width, h = height

Cylinder

2 x pi x r x (r + h)

r = radius, h = height

Cone

pi x r x (r + l), where l = sqrt(r² + h²)

r = base radius, h = height, l = slant height

Square Pyramid

a² + 2 x a x l, where l = sqrt((a/2)² + h²)

a = base side, h = height, l = slant height

Triangular Prism

(b x h_t) + L x (b + s1 + s2)

b = triangle base, h_t = triangle height, L = prism length, s1/s2 = triangle sides

Worked Example: Sphere with Radius 5 cm

Using the formula SA = 4 x pi x r² with r = 5 cm:

r² = 5 x 5 = 25 cm²

4 x pi x 25 = 100 x 3.14159 = 314.16 cm²

The total surface area of a sphere with radius 5 cm is 314.16 cm². For a sphere, the entire surface is curved, so the lateral surface area equals the total surface area.

Total Surface Area vs Lateral Surface Area

Total surface area (TSA) includes every face: the base(s) and all sides. Lateral surface area (LSA) counts only the side faces, leaving out the top and bottom. For shapes like cylinders, cones, and pyramids, the distinction matters when you need to know only the area of the walls (for example, when painting a room you might want only the wall area, not the floor or ceiling).

For a sphere, the entire surface is curved with no base, so TSA = LSA. For a cube or cuboid, the lateral surface area shown counts only the four side (vertical) faces and excludes the top and bottom, so it is smaller than the total surface area.

Practical Applications

Surface area calculations are used in everyday tasks including: estimating how much paint, fabric, or wrapping material is needed to cover a solid object; calculating heat loss through walls or pipes (which is proportional to surface area); working out the amount of material needed to manufacture a product; and packaging and logistics design. In science, surface area is important for understanding rates of reaction, heat transfer, and fluid dynamics.

Related Calculators

Sources and method: Formulae from standard geometry references: Weisstein, E.W. "Surface Area" (MathWorld, Wolfram Research); Coxeter, H.S.M., "Introduction to Geometry" (Wiley, 1989). Slant heights derived from the Pythagorean theorem. Calculations use pi = 3.14159265358979.

This calculator applies standard geometric formulae for regular (perfect) solids. Irregular or composite shapes may require separate calculations for each component. All inputs are assumed to be positive real numbers in the same unit of measurement.

Frequently asked questions

What is the surface area of a sphere?

The surface area of a sphere is 4 x pi x r^2, where r is the radius. For example, a sphere with a radius of 5 cm has a surface area of 4 x 3.14159 x 25 = 314.16 cm squared. This formula gives the total area of the curved surface that encloses the sphere.

What is the difference between total surface area and lateral surface area?

Total surface area (TSA) is the combined area of every face of a solid, including the top, bottom, and all sides. Lateral surface area (LSA) includes only the side faces, excluding the top and bottom bases. For a cylinder, the lateral surface area is 2 x pi x r x h (just the curved wall), while the total surface area adds two circular bases: TSA = 2 x pi x r x (r + h). This calculator shows total surface area for all solids.

How do I find the surface area of a cone?

The total surface area of a cone is pi x r x (r + l), where r is the base radius and l is the slant height. If you know the height (h) instead of the slant height, first calculate l = sqrt(r^2 + h^2). For example, a cone with base radius 3 cm and height 4 cm has slant height sqrt(9 + 16) = 5 cm, and a total surface area of pi x 3 x (3 + 5) = 3.14159 x 24 = 75.40 cm squared.

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