# What is the surface area of a solid?

Finds the total, side (lateral), and base surface area of a cube, rectangular box, sphere, cylinder, cone, or square pyramid from its dimensions.

- Page: https://www.acalculator.org/math/surface-area-calculator
- JSON spec: https://www.acalculator.org/math/surface-area-calculator.json
- Version: 24fd24accc7b

## Default answer

Example with the default inputs (Shape Cylinder, Radius 3, Height 5): Cylinder: total surface area 150.796447.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| shape | Shape | The solid: cube, rectangular box, sphere, cylinder, cone, or square pyramid. |
| a | Edge length | The length of an edge of the cube, or of the square base of the pyramid. |
| l | Length | The length of the rectangular box. |
| w | Width | The width of the rectangular box. |
| r | Radius | The radius of the sphere, or of the circular base. |
| h | Height | The vertical height: of the box, between the ends of the cylinder, or from the base to the tip. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| total | Total surface area | The area of every face and curved surface added together. |
| lateral | Side (lateral) area | The area of the sides only, without the top and bottom faces or the base. |
| base | Base area | The area of one base: the bottom face or circle. |
| slant | Slant height | For a cone or a pyramid: the distance up the side from the edge of the base to the tip. |

## Method

Cube 6a²; box 2(lw + lh + wh); sphere 4πr²; cylinder 2πr² + 2πrh; cone πr² + πrs with s = √(r² + h²); square pyramid a² + 2as with s = √(h² + (a/2)²).

## Assumptions

- Every length is in one unit, and each area is in that unit squared.
- The cylinder and cone are right circular (the tip or top is straight above the centre of the base), and the pyramid is a right pyramid with a square base.
- The total includes every base: both ends of a cylinder, the base of a cone or pyramid, and all six faces of a cube or box.
- Each length must be greater than 0.

## Worked examples

1. shape = cylinder, r = 3, h = 5 gives total = 150.796447, lateral = 94.24778, base = 28.274334. Source: OpenStax Prealgebra 2e, Example 9.53 (150.72 with π ≈ 3.14); hand calculation in content.mdx: 2π × 3² + 2π × 3 × 5 = 48π.
2. shape = box, l = 14, w = 9, h = 17 gives total = 1,034, lateral = 782, base = 126. Source: OpenStax Prealgebra 2e, Example 9.47: 2 × 14 × 17 + 2 × 14 × 9 + 2 × 9 × 17 = 1,034.
3. shape = cube, a = 2.5 gives total = 37.5, lateral = 25, base = 6.25. Source: OpenStax Prealgebra 2e, Example 9.49: 6 × 2.5² = 37.5.
4. shape = sphere, r = 6 gives total = 452.389342. Source: OpenStax Prealgebra 2e, Example 9.51 (452.16 with π ≈ 3.14); hand calculation in content.mdx: 4π × 6² = 144π.
5. shape = cone, r = 3, h = 4 gives slant = 5, lateral = 47.12389, base = 28.274334, total = 75.398224. Source: OpenStax Calculus Volume 2, section 2.4 (lateral area πrs); hand calculation in content.mdx: s = √(3² + 4²) = 5, π × 3 × 5 + π × 3² = 24π.
6. shape = pyramid, a = 6, h = 4 gives slant = 5, lateral = 60, base = 36, total = 96. Source: hand calculation in content.mdx: slant = √(4² + 3²) = 5; 4 triangles of ½ × 6 × 5 = 60; 60 + 6² = 96.

## FAQ

### What is surface area?

The total area of all the outside surfaces of a solid, as if you unfolded it flat and measured the paper. It is in square units, such as in² or cm², because it is an area.

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

The lateral (side) area leaves out the flat ends. For a can, the lateral area is the label (2πrh) and the total adds the top and bottom circles (2πr²). A can with radius 3 cm and height 5 cm has a label of 94.25 cm² and a total of 150.80 cm².

### How do I find the surface area of a cylinder?

Add the two circular ends, 2πr², to the side, 2πrh: A = 2πr² + 2πrh = 2πr(r + h). The side unrolls into a rectangle as tall as the cylinder and as wide as the circumference, 2πr.

### How do I find the surface area of a cone?

Add the base circle, πr², to the side, πrs, where s is the slant height √(r² + h²). A cone with radius 3 and height 4 has s = 5, so A = 9π + 15π = 24π = 75.40.

### How do I find the surface area of a square pyramid?

Add the square base, a², to four triangles. Each triangle has base a and height equal to the slant height s = √(h² + (a/2)²), so the four together are 4 × ½ × a × s = 2as. A pyramid with base 6 and height 4 has s = 5 and A = 36 + 60 = 96.

### What is the difference between surface area and volume?

Surface area measures the outside (how much paint or wrapping it needs) in square units. Volume measures the space inside (how much it holds) in cubic units. The cylinder volume and cone volume calculators find the volume.

### What units should I use?

Type every length in the same unit. The areas are in that unit squared: lengths in feet give square feet, and lengths in centimetres give square centimetres.

## Sources

- OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (rectangular solid, cube, sphere, and cylinder surface areas; Examples 9.47, 9.49, 9.51, 9.53). https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area
- OpenStax, Calculus Volume 2, §2.4 Arc Length of a Curve and Surface Area (lateral surface area of a right circular cone, πrs). https://openstax.org/books/calculus-volume-2/pages/2-4-arc-length-of-a-curve-and-surface-area
- NIST Digital Library of Mathematical Functions, §3.12 Mathematical Constants, equation 3.12.1 (π). https://dlmf.nist.gov/3.12