Prism Volume & Surface
Compute volume V = A_base·h and total surface = 2·A_base + perimeter·h for prisms with regular N-gon base.
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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.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.