Rectangular Prism Calculator
Calculate volume, surface area, lateral area, and space diagonal of any rectangular prism (cuboid). Enter any 3 values and solve for all unknowns.
| Property | Value |
|---|
Rectangular Prism Formulas
Rectangular prism formulas relate volume, surface area, and diagonal through the three dimensions: length (l), width (w), and height (h).
Volume: V = l × w × h
Total Surface Area: S = 2(lw + lh + wh)
Lateral Surface Area: Slat = 2h(l + w)
Top/Bottom Area: Stop = Sbot = lw
Space Diagonal: d = √(l² + w² + h²)
| Known Values | Solve For Height |
|---|---|
| l, w, h | Directly given |
| Surface Area S, l, w | h = (S − 2lw) ÷ (2l + 2w) |
| Volume V, l, w | h = V ÷ (l × w) |
| Diagonal d, l, w | h = √(d² − l² − w²) |
Volume and Surface Area Explained
Volume and surface area of a rectangular prism measure fundamentally different things and are used in different practical contexts.
Volume of a Rectangular Prism
Volume measures the three-dimensional space inside the prism, expressed in cubic units. It tells you how much a box can hold. The formula V = lwh multiplies all three dimensions together. Doubling any one dimension doubles the volume; doubling all three increases volume eightfold.
Total Surface Area
Total surface area is the combined area of all six faces. It is used to determine how much material is needed to wrap or coat a box. The formula S = 2(lw + lh + wh) accounts for three pairs of identical rectangular faces.
Lateral Surface Area
Lateral surface area includes only the four side faces, excluding the top and bottom. It equals 2h(l + w). This is useful when only the sides of a container need to be covered, such as painting the walls of a room without the floor or ceiling.
Space Diagonal of a Rectangular Prism
The space diagonal connects two opposite corners and passes through the interior. It extends Pythagoras into three dimensions: d = √(l² + w² + h²). This calculation is used in packaging to determine whether a long item fits diagonally inside a box.