Triangular Prism
A triangular prism is a 3-D solid with two parallel triangular bases and three rectangular side faces.
Formula
V = \left(\frac{1}{2} \times b \times h_{\triangle}\right) \times \text{length}
Definition
A triangular prism has two triangle-shaped parallel bases and three rectangular lateral faces; looking at it end-on you see a triangle, like a classic tent shape. Volume $V = A_{\triangle} \times l$, where $A_{\triangle}$ is the area of the triangular base and $l$ is the prism's length, and surface area $= 2A_{\triangle} + (\text{perimeter of triangle}) \times l$. It has $5$ faces ($2$ triangles $+$ $3$ rectangles), $9$ edges, and $6$ vertices, satisfying Euler's formula $F + V - E = 2$: $5 + 6 - 9 = 2$. As a polyhedron it is the Cartesian product of a triangle $T$ with a line segment; its dual polyhedron is the triangular dipyramid (two tetrahedra joined at a face).
Example
A ramp shaped like a triangular prism with triangle base $6$ m and height $4$ m, and length $10$ m, has volume $(1/2 \times 6 \times 4) \times 10 = 12 \times 10 = 120$ m$^3$. A right triangle base with legs $3$ cm and $4$ cm (hypotenuse $5$ cm) and prism length $12$ cm gives $V = (1/2)(3)(4)(12) = 72$ cm$^3$ and $SA = 2(6) + (3+4+5)(12) = 12 + 144 = 156$ cm$^2$. The cube can be divided into $3$ congruent triangular prisms (or $5$ or $6$ tetrahedra), illustrating how repeated bisection of a prism produces pyramids.
Key Insight
A triangular prism can always be cut into three pyramids of equal volume, a hands-on proof that the pyramid volume formula $(1/3 \times \text{base} \times \text{height})$ yields one-third the prism volume. In crystallography, the trigonal prism is a crystal form with $6$ faces, and the hexagonal prism (two stacked triangular prisms) is the shape of a honeycomb cell, which minimizes material for a given volume, a natural solution to a packing optimization problem.