Prism Volume & Surface

Volume and surface of a prism

A prism is a solid built from two matching parallel polygon bases with parallelograms joining them. Its volume is V = A_base · h, where A_base is the area of one base and h is the perpendicular distance separating the two. Call it a right prism when the side faces stand square to the bases, and oblique when they lean. A regular prism is one whose base is a regular polygon.

In a right prism the lateral area works out to S_lateral = perimeter · h, and adding both bases gives the total, S_total = S_lateral + 2·A_base. Take a hexagonal base with a 4 cm side (A_base ≈ 41.57 cm², perimeter 24 cm) standing 10 cm tall: that comes out to V ≈ 415.7 cm³ and S_total ≈ 323.1 cm².

Applications

It shows up when you size a rectangular fuel tank, a metal ingot cast in a trapezoidal mould, or a house with a pitched gable roof whose volume feeds a rainwater estimate. The same math handles rectangular packaging, shipping containers and aquaria that don't have square bases.

FAQ

Does the prism need to be right? No. Cavalieri's principle keeps V = A_base·h working for oblique prisms too, provided h is the perpendicular height between the parallel bases.

How do I compute A_base for a regular N-gon of side L? Apply A = (N·L²)/(4·tan(π/N)).

Is the lateral area formula the same for oblique prisms? No. With an oblique prism you take the perimeter of the right cross-section and multiply by the lateral edge length, not by the height.

What is the diagonal of a rectangular prism a×b×c? d = √(a² + b² + c²), which falls out of running Pythagoras twice.