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Triangular Prism — Definition, Formula & Examples

A triangular prism is a three-dimensional shape that has two identical triangular bases connected by three rectangular faces. It has 6 vertices, 9 edges, and 5 faces in total.

A triangular prism is a polyhedron composed of two congruent, parallel triangular faces (the bases) and three lateral faces that are parallelograms. When the lateral edges are perpendicular to the bases, the prism is called a right triangular prism, and each lateral face is a rectangle.

Key Formula

V=B×l=12bh×landSA=2B+P×lV = B \times l = \frac{1}{2} b h \times l \qquad \text{and} \qquad SA = 2B + P \times l
Where:
  • VV = Volume of the triangular prism
  • BB = Area of one triangular base
  • bb = Base length of the triangle
  • hh = Height of the triangle (perpendicular to its base)
  • ll = Length (depth) of the prism, connecting the two bases
  • SASA = Total surface area
  • PP = Perimeter of the triangular base

How It Works

To find the volume of a triangular prism, you calculate the area of one triangular base and multiply it by the length (or height) of the prism. For surface area, you add together the areas of both triangular bases and all three rectangular faces. A common shortcut for surface area is to find the perimeter of the triangle, multiply by the prism's length to get the lateral area, then add twice the base area. Many real-world objects are shaped like triangular prisms — think of a tent, a Toblerone box, or a wedge of cheese.

Worked Example

Problem: A triangular prism has a triangular base with a base of 6 cm and a height of 4 cm. The three sides of the triangle measure 6 cm, 5 cm, and 5 cm. The length of the prism is 10 cm. Find its volume and surface area.
Step 1: Find the base area: Use the triangle area formula with base 6 cm and height 4 cm.
B=12×6×4=12 cm2B = \frac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2
Step 2: Calculate the volume: Multiply the base area by the prism's length.
V=12×10=120 cm3V = 12 \times 10 = 120 \text{ cm}^3
Step 3: Find the perimeter of the triangle: Add all three side lengths of the triangular base.
P=6+5+5=16 cmP = 6 + 5 + 5 = 16 \text{ cm}
Step 4: Calculate the surface area: Add the two triangular bases and the lateral area (perimeter times length).
SA=2(12)+16×10=24+160=184 cm2SA = 2(12) + 16 \times 10 = 24 + 160 = 184 \text{ cm}^2
Answer: The volume is 120 cm³ and the surface area is 184 cm².

Another Example

Problem: A camping tent is shaped like a right triangular prism. The triangular opening has a base of 8 ft and a height of 3 ft. The tent is 10 ft long. How much space (volume) is inside the tent?
Step 1: Find the triangular base area: Calculate the area of the triangular cross-section.
B=12×8×3=12 ft2B = \frac{1}{2} \times 8 \times 3 = 12 \text{ ft}^2
Step 2: Multiply by the length: The prism extends 10 ft, so multiply the base area by 10.
V=12×10=120 ft3V = 12 \times 10 = 120 \text{ ft}^3
Answer: The tent encloses 120 ft³ of space.

Visualization

Why It Matters

Triangular prisms appear throughout middle-school geometry courses when students learn to compute volume and surface area of 3D shapes. Engineers and architects use triangular prism calculations when designing roof structures, ramps, and bridge supports. Mastering this shape also builds a foundation for working with more complex prisms and cross-sectional analysis in high-school geometry.

Common Mistakes

Mistake: Confusing the triangle's height with the prism's length
Correction: The triangle's height is the perpendicular distance inside the triangular base, while the prism's length is the distance between the two triangular faces. Label your diagram carefully to keep them separate.
Mistake: Forgetting to include both triangular bases in the surface area
Correction: A triangular prism has two triangular faces, not one. Always multiply the base area by 2 before adding the lateral area.

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