Last updated: 2026-05-18

The rectangular prism

A rectangular prism — also called a cuboid or box — is a three-dimensional shape with six rectangular faces. All angles are right angles, and opposite faces are identical. This calculator computes volume, total surface area, individual face areas, and the space diagonal from any three input dimensions.

Formulas

With length $l$, width $w$, and height $h$:

Volume — the amount of space enclosed:

Total surface area — the sum of all six face areas:

The six faces come in three identical pairs. Each pair contributes twice its area:

Space diagonal — the longest straight line inside the prism, connecting opposite corners:

This is the three-dimensional extension of the Pythagorean theorem: applying $a^2 + b^2 = c^2$ to the base gives the face diagonal $\sqrt{l^2 + w^2}$, and applying the theorem again vertically gives $\sqrt{l^2 + w^2 + h^2}$.

Why the surface area formula works

Every rectangular prism has exactly six faces. Because opposite faces are identical, they pair up:

• Two faces of area $lw$ → contributes $2lw$
• Two faces of area $lh$ → contributes $2lh$
• Two faces of area $wh$ → contributes $2wh$

Total: $SA = 2lw + 2lh + 2wh = 2(lw + lh + wh)$

The factor of 2 is therefore not arbitrary; it counts each face exactly once across the three matching pairs.

Worked example

A standard shipping box measures 38 cm × 28 cm × 22 cm.

Volume: $V = 38 \times 28 \times 22 = 23{,}408 \text{ cm}^3 = 23.4 \text{ L}$

Surface area: $SA = 2(38 \times 28 + 38 \times 22 + 28 \times 22) = 2(1064 + 836 + 616) = 5032 \text{ cm}^2$

The surface area corresponds to roughly 5032 cm² of cardboard required to construct the box, before accounting for flaps and overlap.

Space diagonal: $d = \sqrt{38^2 + 28^2 + 22^2} = \sqrt{1444 + 784 + 484} = \sqrt{2712} \approx 52.1 \text{ cm}$

A pole or diagonal brace fitting inside this box can be at most about 52 cm long.

Practical applications

Shipping and logistics — volume determines how many units fit in a container; surface area determines how much wrapping material is needed. A pallet of boxes is filled by dividing the container volume by the box volume (ignoring gaps).

Aquariums and tanks — volume gives the water capacity; surface area gives the glass area. A 60 cm × 30 cm × 36 cm tank holds $60 \times 30 \times 36 = 64{,}800 \text{ cm}^3 = 64.8 \text{ L}$.

Construction and woodworking — a timber beam is a rectangular prism; its volume (in m³) determines the weight (density × volume). Knowing face areas helps estimate paint or finish coverage.

Room dimensions — a room is approximately a rectangular prism. Its volume determines heating/cooling load; floor area ($l \times w$) determines floor material needs.

Rectangular prism vs. cube

A cube is the special case where $l = w = h$. The formulas simplify to $V = l^3$ and $SA = 6l^2$. Any rectangular prism with at least one unequal dimension is a cuboid but not a cube.