Triangular prism

Properties

• 5 faces (2 triangles and 3 rectangles), 9 edges and 6 corners.
• Every slice parallel to the ends is the same triangle.
• A cuboid is a prism with rectangular ends, and a cylinder behaves like a prism with circular ends.

Why the formulas work

For every prism, the volume is the area of the end multiplied by the length, because the same cross-section is repeated all the way along. The three rectangular faces, unfolded, make one rectangle whose width is the perimeter of the triangle and whose height is the length of the prism.

Worked example

Take a triangular prism with side a of the triangle 3 cm, side b 4 cm, side c 5 cm and length of the prism 10 cm. First, half the perimeter s = 6.

1. Area of one end. End area by Heron's formula. Here: √(6 × (6 − 3) × (6 − 4) × (6 − 5)) = 6 cm².
2. Volume. Volume = end area × L. Here: 6 × 10 = 60 cm³.
3. Lateral surface area. Lateral surface area = (a + b + c) × L. Here: (3 + 4 + 5) × 10 = 120 cm².
4. Total surface area. Total surface area = lateral area + 2 × end area. Here: 120 + 2 × 6 = 132 cm².

Finding a missing measurement

Length = volume ÷ area of the end. A prism of volume 60 cm³ whose end is a triangle of sides 3 cm, 4 cm and 5 cm, with area 6 cm², is 10 cm long. When the end is a right-angled triangle, its area is simply half the product of the two shorter sides.

Mistakes to avoid

• Using the triangle's height as the prism's length, or the other way round.
• Leaving out the two triangular ends when the total surface is asked.

Where you see it

A tent, a triangular chocolate bar box, a glass prism used to split light, a wedge and the roof space of a house are triangular prisms.

Try these

Work each one out, then tap to check.

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