Prism
A 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, parallel polygonal bases connected by rectangular (or, for oblique prisms, parallelogram) side faces; volume $= \text{base area} \times \text{height}$, and surface area $= \text{lateral area} + 2 \times \text{base area}$. Formally, a prism is the Cartesian product of a polygon $P$ with a line segment $[0, h]$, $\{(x, y, z) : (x, y) \in P, 0 \le z \le h\}$, so its volume is $\text{Area}(P) \cdot h$ by Cavalieri's principle, since every cross-section parallel to the base is congruent to $P$; oblique prisms use the same formula with $h$ as perpendicular height.
Example
A box of crackers is a rectangular prism, and a Toblerone chocolate bar box is a triangular prism, both with two matching ends and flat sides connecting them. A pentagonal prism has $2$ pentagon bases and $5$ rectangular sides, giving $7$ faces, $15$ edges, and $10$ vertices, satisfying Euler's formula $F + V - E = 2$: $7 + 10 - 15 = 2$. 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, a formula that generalizes to pyramids and cones.
Key Insight
Prisms are named after their base shape, a triangular prism has triangle bases and a hexagonal prism has hexagon bases; the word "prism" in optics refers to the triangular glass prism that splits white light into a rainbow. Every prism satisfies Euler's polyhedral formula $F + V - E = 2$: for n-gon bases, $F = n + 2$, $V = 2n$, $E = 3n$, giving $(n+2) + 2n - 3n = 2$. The prismatoid volume formula $V = (h/6)(A_1 + 4A_m + A_2)$ is Simpson's rule applied to the area function, a deep connection between solid geometry and numerical integration.