Surface Area Calculator — Step-by-Step Solutions

The Surface Area Calculator computes the total surface area of common 3D shapes. Select a shape, enter its dimensions, and get the result instantly with step-by-step working showing each formula.

Supported shapes include cubes, rectangular prisms, spheres, cylinders, cones, and triangular prisms. All standard geometry formulas are applied and explained.

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How to Use the Surface Area Calculator

Select a 3D shape from the mode buttons: Cube, Rectangular Prism, Sphere, Cylinder, Cone, or Triangular Prism.
Enter the required dimensions for your chosen shape. Each shape requires different inputs (side length, radius, height, etc.).
The surface area is calculated instantly as you type. Results appear below the inputs with full step-by-step working.
The step-by-step breakdown shows the formula used, substitution of your values, and intermediate calculations leading to the final answer.

Formula and Theory

Surface area is the total area of all the faces (or surfaces) of a three-dimensional object. It is measured in square units (e.g. cm squared, m squared).

For a cube with side length a, every face is an identical square. Since a cube has 6 faces, the surface area is SA = 6a squared. A rectangular prism generalises this: SA = 2(lw + lh + wh), accounting for three pairs of rectangular faces.

For a sphere, the surface area formula SA = 4 pi r squared was first proved by Archimedes. A cylinder combines two circular ends (2 pi r squared) with a lateral surface that unwraps into a rectangle (2 pi r h), giving SA = 2 pi r squared + 2 pi r h.

A cone's surface area includes the circular base (pi r squared) and a lateral surface that unwraps into a sector. The lateral area depends on the slant height l = sqrt(r squared + h squared), giving a lateral area of pi r l. A triangular prism has two triangular faces and three rectangular faces, so SA = 2 times the triangle area plus the perimeter times the prism length.

Worked Examples

Example 1: Surface area of a cube

Problem: Find the surface area of a cube with side length 5 cm.

Solution: SA = 6a squared = 6 x 5 squared = 6 x 25 = 150.

Answer: 150 cm squared

Example 2: Surface area of a sphere

Problem: Find the surface area of a sphere with radius 7 cm.

Solution: SA = 4 pi r squared = 4 x pi x 49 = 196 pi = 615.75.

Answer: 615.75 cm squared

Example 3: Surface area of a cylinder

Problem: Find the surface area of a cylinder with radius 3 cm and height 10 cm.

Solution: SA = 2 pi r squared + 2 pi r h = 2 pi (9) + 2 pi (30) = 18 pi + 60 pi = 78 pi = 245.04.

Answer: 245.04 cm squared

Example 4: Surface area of a cone

Problem: Find the surface area of a cone with radius 4 cm and height 6 cm.

Solution: Slant height l = sqrt(16 + 36) = sqrt(52) = 7.211. SA = pi r squared + pi r l = 16 pi + 4 pi (7.211) = 16 pi + 28.844 pi = 44.844 pi = 140.88.

Answer: 140.88 cm squared

Frequently Asked Questions

What is the difference between surface area and volume?

Surface area measures the total area of the outside faces of a 3D shape (in square units), while volume measures the space enclosed inside the shape (in cubic units). They use different formulas and have different units.

When would I need to calculate surface area?

Surface area calculations are used when wrapping or covering objects (paint, fabric, packaging), calculating heat transfer, determining material costs for manufacturing, and solving geometry problems in maths and physics.

How do I find the surface area of an irregular shape?

For irregular shapes, break them down into simpler components (cubes, cylinders, etc.), calculate each surface area separately, then add them together. Subtract any areas where shapes join and the surface is internal.

What is lateral surface area?

Lateral surface area is the area of just the sides of a 3D shape, excluding the top and bottom (or base) faces. For a cylinder, the lateral surface area is 2 pi r h.