Rectangular Prism Volume: The Formula and a Worked Example

Right Rectangular Prism: Definition

Wolfram MathWorld defines a cuboid — also called a rectangular parallelepiped — as “a closed box composed of three pairs of rectangular faces placed opposite each other and joined at right angles to each other.” Math Is Fun describes the same shape more plainly: a box-shaped object with six flat faces, all of them rectangles, where every angle is a right angle. That right-angle property is why the shape is also called a right rectangular prism: its side faces meet the base square on, not at a slant.

The Enciclopedia della Matematica Treccani places it inside the broader family of parallelepipeds — solids with six faces, opposite pairs parallel and congruent — and calls one “rectangular” only when every face is a rectangle. Two other names turn up for the exact same solid in textbooks: quadrangular rectangular prism and rectangular-based parallelepiped.

The Rectangular Prism Formula for Volume

Every source above gives the same rectangular prism formula, just with different letters. Treccani and Wolfram MathWorld write it as V = abc, where a, b and c are the three edges; Math Is Fun and Cuemath write it as V = lwh, length times width times height. Same multiplication, different notation:

V = length × width × height

The three dimensions meet at each corner of the solid at right angles to one another. The result comes out in cubic units: if length, width and height are measured in centimeters, the volume is in cubic centimeters.

The same formula also reads as “base area × height,” the rule behind the triangular-prism case too — there, though, finding the base area takes the triangle-area formula first, since a plain product of two sides isn’t enough. In a rectangular prism the base is itself a rectangle, so its area is already length × width, and multiplying that by the height lands on the same number as V = l × w × h.

How to Find the Volume of a Rectangular Prism: Worked Example

Practical example: Cuemath works through a box measuring 8 inches long, 5 inches wide and 16 inches tall. First, the base area: 8 × 5 = 40 square inches. Then multiply by the height: 40 × 16 = 640 cubic inches. Multiplying all three numbers in one go, 8 × 5 × 16, gives the same 640.

The Cube: A Special Case

Treccani and Wolfram MathWorld both mark the same boundary: once the three dimensions match (a = b = c), a rectangular prism turns into a cube. A cube, then, is simply the special case where length, width and height happen to be equal, so the volume formula stays exactly the same, just with three matching factors (5 × 5 × 5, for instance).

Total Surface Area, So It Doesn’t Get Mixed Up with Volume

The same sources also give the formula for total surface area, the combined area of all six faces, mostly so it doesn’t get confused with volume: A = 2(lw + lh + wh), using the same three dimensions. That’s a different formula from the one above, and the result comes out in square units: for a solid measuring 4, 5 and 10, the volume is 200 cubic units, while the total surface area is 220 square units. The algebra behind these formulas is covered in algebra: what it is and how symbolic notation works. Both formulas rest on the same theorems as Euclidean geometry: what it is and how Euclid’s postulates work, and the broader framework for combining variables and matrices is linear algebra: what it is and how vectors and matrices work.