# How big is my rectangular prism?

Finds the volume, total surface area, lateral area, base area and space diagonal of a rectangular prism (a box) from its length, width and height.

- Page: https://www.acalculator.org/math/rectangular-prism-calculator
- JSON spec: https://www.acalculator.org/math/rectangular-prism-calculator.json
- Version: a5f1685fc571

## Default answer

Example with the default inputs (Length 10, Width 5, Height 3): A box 10 × 5 × 3 has a volume of 150 and a surface area of 190.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| l | Length | The length of the base of the box. |
| w | Width | The width of the base of the box. |
| h | Height | The height of the box, at right angles to the base. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| volume | Volume | Length × width × height, in cubic units. |
| surface | Total surface area | All six faces: 2(lw + lh + wh), in square units. |
| lateral | Lateral area | The four side faces, without top and bottom: 2h(l + w). |
| base | Base area | The bottom face: length × width. |
| diagonal | Space diagonal | The distance from one corner to the opposite corner: √(l² + w² + h²). |

## Method

V = l × w × h; SA = 2(lw + lh + wh); lateral area = 2h(l + w); base area = l × w; diagonal = √(l² + w² + h²).

## Assumptions

- The prism is a right rectangular prism (a box): every face is a rectangle and every corner is a right angle.
- All lengths are in the same unit; areas come out in that unit squared and the volume in that unit cubed.

## Worked examples

1. l = 5, w = 10, h = 3 gives surface = 190, volume = 150, lateral = 90, base = 50, diagonal = 11.575837. Source: OpenStax, Contemporary Mathematics, §10.7 Volume and Surface Area (right prism SA = 2B + ph, V = B·h; Example 10.56), https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area (SA = 2(10)(5) + 2(5)(3) + 2(10)(3) = 190 cm², V = 150 cm³); OpenStax, Calculus Volume 3, §2.2 Vectors in Three Dimensions (distance between two points in space), https://openstax.org/books/calculus-volume-3/pages/2-2-vectors-in-three-dimensions.
2. l = 15, w = 6, h = 6 gives surface = 432, volume = 540, lateral = 252, diagonal = 17.233688. Source: OpenStax, Contemporary Mathematics, §10.7 Volume and Surface Area (right prism SA = 2B + ph, V = B·h; Example 10.56), https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area (Your Turn 10.56).
3. l = 2, w = 3, h = 6 gives volume = 36, surface = 72, diagonal = 7. Source: OpenStax, Contemporary Mathematics, §10.7 Volume and Surface Area (right prism SA = 2B + ph, V = B·h; Example 10.56), https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area; OpenStax, Calculus Volume 3, §2.2 Vectors in Three Dimensions (distance between two points in space), https://openstax.org/books/calculus-volume-3/pages/2-2-vectors-in-three-dimensions.
4. l = 0.1, w = 0.2, h = 0.3 gives volume = 0.006, surface = 0.22, base = 0.02. Source: OpenStax, Contemporary Mathematics, §10.7 Volume and Surface Area (right prism SA = 2B + ph, V = B·h; Example 10.56), https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area.

## FAQ

### How do I find the volume of a rectangular prism?

Multiply the three edge lengths: V = l × w × h. A box 5 cm long, 10 cm wide and 3 cm high holds 5 × 10 × 3 = 150 cm³.

### How do I find the surface area of a rectangular prism?

Add the areas of the six faces, which come in three equal pairs: SA = 2(lw + lh + wh). For the 5 × 10 × 3 box that is 2(50 + 15 + 30) = 190 cm².

### What is the lateral area?

The area of the four side faces only, without the top and the bottom: 2h(l + w). It is the area of the wrapping around a box with no lid and no floor, such as the walls of a room.

### How long is the diagonal of a box?

The space diagonal runs from one corner through the inside to the opposite corner: d = √(l² + w² + h²). A 2 × 3 × 6 box has a diagonal of √49 = 7.

### What units does the calculator use?

Any one unit for all three lengths. Areas come out in that unit squared and the volume in that unit cubed, so inches give in² and in³ and metres give m² and m³.

### Is a cube a rectangular prism?

Yes. A cube is a rectangular prism with three equal edges s, so V = s³, SA = 6s² and the diagonal is s√3.

## Sources

- OpenStax, Contemporary Mathematics, §10.7 Volume and Surface Area: right prism SA = 2B + ph and V = B·h; Example 10.56 (width 10 cm, length 5 cm, height 3 cm: SA = 190 cm², V = 150 cm³) and Your Turn 10.56. https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area (retrieved 2026-10-02)
- OpenStax, Calculus Volume 3, §2.2 Vectors in Three Dimensions: the distance between two points in space, which gives the space diagonal. https://openstax.org/books/calculus-volume-3/pages/2-2-vectors-in-three-dimensions (retrieved 2026-10-02)