Rectangular Prism and Cube Calculator

How it works

A rectangular prism, or box, has six rectangular faces meeting at right angles. Its volume is simply length times width times height. Its surface area is the sum of the six faces, which come in three matching pairs: two of length by width, two of length by height and two of width by height. The space diagonal runs from one corner to the opposite corner through the inside of the box, and the Pythagorean theorem applied twice gives it as the square root of the three edges squared and added.

A cube is the special case where all three edges are the same length a, so the volume is a cubed, the surface area is six squares of a squared, and the diagonal is a times the square root of three. Because each of those is a one-line function of a, the calculator can also run them backwards: the cube root of a volume, or the square root of a sixth of a surface area, recovers the edge.

The unit you pick only labels the lengths, areas and volumes; the geometry is the same whatever the unit. The capacity line is where the unit matters: the volume in cubic inches, feet, yards, centimetres or metres is converted to US gallons and litres using the legal definitions of those units, so a one-foot cube reports 7.48 gallons and a one-metre cube 1,000 litres.

Formula

V  = l × w × h
SA = 2(lw + lh + wh)
d  = √(l² + w² + h²)
cube: V = a³   SA = 6a²   d = a√3   a = ∛V = √(SA ÷ 6)
capacity: 1 US gal = 231 in³ = 3.785411784 L;  1 ft³ = 7.48052 gal;  1 m³ = 1,000 L

Example

A box 4 ft long, 3 ft wide and 2 ft high has a volume of 4 × 3 × 2 = 24 ft³, a surface area of 2 × (12 + 6 + 8) = 52 ft², and a space diagonal of √(16 + 9 + 4) = √29 ≈ 5.3852 ft. Filled with water it would hold 24 × 7.48052 ≈ 179.53 US gallons, or 679.60 litres.

A cube with 1 ft edges has a volume of exactly 1 ft³, which is 7.48 US gallons or 28.32 litres, a surface area of 6 ft², and a diagonal of √3 ≈ 1.7321 ft. Running it the other way, a cube whose volume is 27 ft³ has edges of ∛27 = 3 ft and a surface area of 6 × 9 = 54 ft².

Assumptions and limitations

• Every face meets its neighbours at right angles, so the solid is a true box. A skewed box (a parallelepiped) has the same volume but a different surface area and diagonal.
• All edges are in the unit you chose, and the capacity assumes the whole interior volume is usable. A real tank, aquarium or crate holds less once wall thickness, headspace and fittings are taken out.
• The gallon is the US liquid gallon of 231 cubic inches, not the imperial gallon.
• Results are rounded for display only.

Frequently asked questions