Prism
Geometry & MeasurementA prism is a 3-D solid with two parallel, congruent polygon bases connected by rectangular side faces.
Formula
V = B \times h \ (B = \text{base area})
Definition
A prism is a 3-D shape with two matching flat ends (bases) that are the same shape and size, connected by flat rectangular sides. The bases are parallel to each other.
Example
A box of crackers is a rectangular prism. A Toblerone chocolate bar box is a triangular prism. Both have two matching ends and flat sides connecting them.
Key Insight
Prisms are named after their base shape. A prism with triangle bases is a triangular prism; one with hexagon bases is a hexagonal prism. The word "prism" in optics refers to the triangular glass prism that splits white light into a rainbow.
Definition
A right prism has two congruent, parallel polygonal bases connected by lateral faces that are rectangles perpendicular to the bases. An oblique prism has lateral faces that are parallelograms. Volume $= \text{base area} \times \text{height}$; Surface area $= \text{lateral area} + 2 \times \text{base area}$.
Example
A pentagonal prism has $2$ pentagon bases and $5$ rectangular sides, giving $7$ faces, $15$ edges, and $10$ vertices. By Euler's formula $F + V - E = 2$: $7 + 10 - 15 = 2$. Volume = area of pentagon $\times$ height.
Key Insight
Every prism satisfies Euler's polyhedral formula $F + V - E = 2$. For a prism with n-gon bases: $F = n + 2$, $V = 2n$, $E = 3n$, giving $(n+2) + 2n - 3n = 2$. This formula connects the geometry of any convex polyhedron.
Definition
A prism is the Cartesian product of a polygon $P$ with a line segment $[0, h]$: formally, $\{(x, y, z) : (x, y) \in P, 0 \le z \le h\}$. Its volume is $\text{Area}(P) \cdot h$ by Cavalieri's principle (every cross-section parallel to the base is congruent to $P$). Oblique prisms have the same volume formula with $h$ as perpendicular height.
Example
A truncated prism (frustum of a prism) can be computed by the prismatoid formula $V = (h/6)(A_{top} + A_{bottom} + 4A_{mid})$, where $A_{mid}$ is the area of the cross-section at mid-height. This generalizes to pyramids and cones.
Key Insight
The prismatoid volume formula $V = (h/6)(A_1 + 4A_m + A_2)$ (Simpson's rule applied to the area function) reveals a deep connection between solid geometry and numerical integration, showing that Simpson's rule is exact for quadratic functions.