A surface area calculator finds the total outside area of a cube, sphere, cylinder or rectangular box. A 4 × 3 × 2 box has a surface area of 52 square units.
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How to use the surface area calculator
- Choose the shape.
- Enter the dimensions in one unit.
- Read the surface area in square units.
How the surface area calculator works
The surface area calculator finds the total outside area of four common solids. Choose a shape and enter its dimensions in any one unit; the result is in that unit squared. The formulas are:
- Cube: 6a²
- Sphere: 4πr²
- Cylinder: 2πr(r + h)
- Rectangular box: 2(lw + lh + wh)
Results show up to 4 decimal places.
Surface area calculator worked example
A box is 4 by 3 by 2 inches. The three pairs of faces are 4 × 3 = 12, 4 × 2 = 8 and 3 × 2 = 6. Adding them gives 26, and doubling gives 52 square inches. A cylinder with radius 2 and height 5 has 2π × 2 × 7 = 28π ≈ 87.9646 square units: two circular ends of 4π each plus a curved side of 20π. The surface area calculator does both in one step.
| Shape | Dimensions | Surface area |
|---|---|---|
| Cube | side 4 | 96 |
| Sphere | r 3 | 113.0973 |
| Cylinder | r 2, h 5 | 87.9646 |
| Box | 4 × 3 × 2 | 52 |
| Box | 10 × 8 × 6 | 376 |
| Sphere | r 1 | 12.5664 |
Surface area versus volume
Surface area is measured in square units and tells you how much material covers the outside. Volume is in cubic units and tells you how much fits inside. Doubling every dimension multiplies surface area by 4 and volume by 8. That is why small objects have more surface per unit of volume, which matters for heat loss and chemical reactions.
Practical uses
Surface area tells you how much paint, wrapping, insulation or sheet metal you need. To paint a box-shaped shed, find the area of the walls and roof and subtract doors and windows. For a cylindrical tank with an open top, use πr² + 2πrh instead of the full formula, since one end is missing. Add 10% or so for waste and overlap.
Units
Keep every dimension in the same unit. If the result is in square inches, divide by 144 for square feet; square centimetres divide by 10,000 for square metres. Mixing feet and inches is the most common mistake.
Other shapes
A cone has surface area πr² + πrs, where s is the slant height. A square pyramid with base side b and slant height s has b² + 2bs. For a hemisphere including its flat base, use 3πr². Split complex objects into simple parts, add the outer surfaces and leave out faces that are joined together.
Entering dimensions
Enter positive values only. The tool rejects zero and negative dimensions, because a negative radius squared would still give a positive area with no physical meaning. Use the same unit for every dimension.
Why the sphere formula works
The surface area calculator uses a formula first found by Archimedes. He showed that a sphere has the same surface area as the curved side of the cylinder that just encloses it. That cylinder has radius r and height 2r, so its side is 2πr × 2r = 4πr². A ball with radius 3 therefore has a surface area of 36π, about 113.1 square units, and a ball with radius 1 has about 12.57.
Using a net
A net is the flat pattern you get by unfolding a solid. The net of a box is six rectangles in three matching pairs; the net of a cylinder is two circles and one rectangle whose length is the circumference 2πr. Drawing a net is a good way to check that you have counted every face exactly once, and it is how packaging and sheet-metal parts are designed. Compare your hand total with the surface area calculator result.
Tips
- Use one unit for all dimensions.
- Remember that the result is in square units.
- Remove faces that are open or hidden.
- Add a margin for waste when buying material.
Related converters and calculators
Related calculators:
- Volume calculator: capacity
- Area calculator: flat shapes
- Square footage calculator: rooms
- Roofing calculator: roof area
Sources
OpenStax Prealgebra 2e, 9.6: Volume and surface area; NIST: SI units – length.
Questions about this calculator
What is the cube formula?
6a².
What is the sphere formula?
4πr².
What is the cylinder formula?
2πr(r + h).
What is the box formula?
2(lw + lh + wh).
What units is the result in?
Your unit squared.
Check the assumptions and units before relying on a result. Financial, tax and health estimates do not replace advice for your individual circumstances.
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