Triangular Prism
Geometry & MeasurementA 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 ends and three flat rectangular sides. If you look at it from the end, you see a triangle. A classic tent shape is a triangular prism.
Example
A ramp shaped like a triangular prism: the triangle end has base $6$ m and height $4$ m, and the ramp is $10$ m long. Volume $= (1/2 \times 6 \times 4) \times 10 = 12 \times 10 = 120$ m$^3$.
Key Insight
A triangular prism has $5$ faces ($2$ triangles $+$ $3$ rectangles), $9$ edges, and $6$ vertices. Check with Euler's formula: $5 + 6 - 9 = 2$. It always works for convex polyhedra.
Definition
A triangular prism has $2$ congruent parallel triangle bases and $3$ rectangular lateral faces. Volume $V = A_{\triangle} \times l$, where $A_{\triangle}$ is the area of the triangular base and $l$ is the prism's length. Surface area $= 2A_{\triangle} + (\text{perimeter of triangle}) \times l$.
Example
Base triangle: right triangle with legs $3$ cm and $4$ cm (hypotenuse $5$ cm). Prism length $12$ cm. $V = (1/2)(3)(4)(12) = 72$ cm$^3$. $SA = 2(6) + (3+4+5)(12) = 12 + 144 = 156$ cm$^2$.
Key Insight
A triangular prism can always be cut into three pyramids of equal volume. This gives a hands-on proof that the pyramid volume formula $(1/3 \times \text{base} \times \text{height})$ yields one-third the prism volume, since each pyramid has volume $V_{prism}/3$.
Definition
A triangular prism is a polyhedron with $f=5$, $v=6$, $e=9$ satisfying Euler's formula. 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). A triangular prism is also a special case of a wedge and appears in the decomposition of the cube into $5$ tetrahedra.
Example
The cube can be divided into $5$ tetrahedra (or $6$ equal tetrahedra, or $3$ congruent triangular prisms). The three-prism decomposition illustrates how repeated bisection of a prism produces pyramids, connecting prism and pyramid volume formulas.
Key Insight
In crystallography, the trigonal prism is a crystal form with $6$ faces. 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.