acalculator

How big is my rectangular prism?

Type the length, width and height of the box. The rectangular prism calculator gives the volume, the surface area of all six faces, the area of the four sides, and the corner-to-corner diagonal.

Your numbers

Volume
150

A box 10 × 5 × 3 has a volume of 150 and a surface area of 190.

Total surface area
190
Lateral area
90
Base area
50
Space diagonal
11.5758

Volume: 150. A box 10 × 5 × 3 has a volume of 150 and a surface area of 190.

What does the base look like?

How to calculate

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.

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.

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

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Length 5, Width 10, Height 3 gives Total surface area 190, Volume 150, Lateral area 90, Base area 50, Space 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. Length 15, Width 6, Height 6 gives Total surface area 432, Volume 540, Lateral area 252, Space 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. Length 2, Width 3, Height 6 gives Volume 36, Total surface area 72, Space 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. Length 0.1, Width 0.2, Height 0.3 gives Volume 0.006, Total surface area 0.22, Base area 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

How it works

A rectangular prism has a rectangular base of length l and width w, and a height h at right angles to the base.

  • Volume V = l × w × h.
  • Base area B = l × w.
  • Lateral area = 2h(l + w): the four side faces (the perimeter of the base times the height).
  • Total surface area SA = 2(lw + lh + wh) = 2B + lateral area.
  • Space diagonal d = √(l² + w² + h²): the straight line from a bottom corner to the opposite top corner.

Rules. Each length is more than 0 and at most 1,000,000.

Exact arithmetic. Each length is read as the exact decimal you typed (0.1 is 1/10), so the volume and the areas are exact fractions. The diagonal is exact when l² + w² + h² is the square of a fraction (√49 = 7), and otherwise a double-precision square root.

Output format. Every result shows at most 4 decimals, rounded half up from its decimal value; trailing zeros are dropped. Areas are in square units and the volume in cubic units of the length unit you used.

Assumptions

  • The prism is a right rectangular prism: all faces are rectangles and all corners are right angles.
  • All three lengths use the same unit.

Worked examples by hand

Length 5, width 10, height 3 (OpenStax Example 10.56). B = 5 × 10 = 50. V = 50 × 3 = 150. Lateral area = 2 × 3 × (5 + 10) = 90. SA = 2(50 + 15 + 30) = 190. d = √(25 + 100 + 9) = √134 = 11.5758.

Length 15, width 6, height 6 (OpenStax Your Turn 10.56). V = 15 × 6 × 6 = 540. Lateral area = 2 × 6 × 21 = 252. SA = 2(90 + 90 + 36) = 432. d = √(225 + 36 + 36) = √297 = 17.2337.

Length 2, width 3, height 6. V = 36. SA = 2(6 + 12 + 18) = 72. d = √(4 + 9 + 36) = √49 = 7.

Length 0.1, width 0.2, height 0.3. B = 0.02; V = 0.02 × 0.3 = 0.006; SA = 2(0.02 + 0.03 + 0.06) = 0.22.

Other questions people ask

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.