Volume of Prisms

Pre-Algebra · Axiom Academy

Why the space inside any prism is just its base area, copied straight up: V = base area × height. Take the flat base and make many thin copies of it. Stack those copies straight up, one on top of the next, until the pile is as tall as the height . The copies fill the prism exactly — so the volume is just the area of one copy, added up once for every layer of height. Every layer has the same area as the base, B The height h counts how many layers stack up The stacking picture never asked what the base looked like — only its area . So swap the base for a rectangle, a triangle, or a hexagon and extrude each one up by the same height: the rule does not change. Find the area of the base, multiply by the height, done. B is the area of the base; h is the perpendicular height between the two bases. The only part that changes from prism to prism is how you find B . A rectangle's base area is length × width; a triangle's is ½ × base × height of that triangle; a regular hexagon's has its own formula. Whatever the shape, once you know B you multiply by h . Watch the rule run on a real box: shade the base to get its area, then stack that area up through the height. The running count of cubic units lands on the volume. A box has length 6 cm, width 4 cm, and height 5 cm. Its base is a rectangle, so the base area is length × width. Stacking the base 24 cm² up through 5 cm of height gives 120 cubic centimeters. Example 2 — a cube (a special prism)

This is the written version of the interactive lesson above. See the full Pre-Algebra course.