Right prisms and right pyramids with triangular or square bases

Learning outcome

Calculate prism and pyramid areas and volumes, distinguishing perpendicular height from face slant height.

Concepts and assumptions

A right prism has parallel, congruent bases and perpendicular lateral edges. With base area B, perimeter p and height h, identical layers give volume B × h. Rectangular walls give lateral area p × h; total area = p × h + 2 × B.

A pyramid’s tapering sections give volume B × h ÷ 3, one-third of the matching prism. Height h is perpendicular to the base plane, not an altitude within the base triangle.

Here pyramids have square or equilateral bases, with the apex directly above their centre. Equal face altitudes s give lateral area p × s ÷ 2; total area adds B. If centre-to-side distance is ρ, s² = h² + ρ². The apex-to-vertex edge is different.

For arbitrary triangular bases, face altitudes may differ: add their individual triangular areas instead. Dimensions are exact centimetres; leave radicals exact without rounding.

Worked examples

Example 1 — Triangular prism. The base is a right triangle with perpendicular sides 6 cm and 8 cm and hypotenuse 10 cm; prism height is 15 cm. B = 6 × 8 ÷ 2 = 24 cm²; p = 24 cm. Volume = 24 × 15 = 360 cm³. Total area = 24 × 15 + 2 × 24 = 408 cm².

Example 2 — Square pyramid. Base side is 10 cm and perpendicular height is 12 cm. Centre-to-side distance = 5 cm, so s = √(144 + 25) = 13 cm. Lateral area = 40 × 13 ÷ 2 = 260 cm². Total area = 260 + 100 = 360 cm²; volume = 100 × 12 ÷ 3 = 400 cm³.

Example 3 — Regular triangular pyramid. Its base has side 6√3 cm, area 27√3 cm² and centre-to-side distance 3 cm. Perpendicular height is 4 cm. Thus s = √(16 + 9) = 5 cm. Base perimeter = 18√3 cm. Lateral area = 18√3 × 5 ÷ 2 = 45√3 cm²; total area = 45√3 + 27√3 = 72√3 cm². Volume = 27√3 × 4 ÷ 3 = 36√3 cm³.

Common mistakes

A triangular prism has rectangular walls, unlike a pyramid. A triangular base alone does not guarantee equal face slant heights.

Practice questions

1. A right prism has a right-triangular base with sides 5, 12 and 13 cm, and height 10 cm. Find total area and volume.
2. A right prism has square base side 7 cm and height 12 cm. Find total area and volume.
3. A right square pyramid has base side 12 cm and perpendicular height 8 cm. Find total area and volume.
4. A regular triangular pyramid has equilateral base side 8 cm and face slant height 10 cm. Find lateral and total areas.

Worked solutions

1. B = 5 × 12 ÷ 2 = 30 cm²; p = 30 cm. Total area = 30 × 10 + 60 = 360 cm²; volume = 30 × 10 = 300 cm³.
2. B = 49 cm²; p = 28 cm. Total area = 28 × 12 + 98 = 434 cm²; volume = 49 × 12 = 588 cm³.
3. s = √(8² + 6²) = 10 cm. Total area = 48 × 10 ÷ 2 + 144 = 384 cm²; volume = 144 × 8 ÷ 3 = 384 cm³.
4. Base altitude = √(64 - 16) = 4√3 cm, so B = 8 × 4√3 ÷ 2 = 16√3 cm². Lateral area = 24 × 10 ÷ 2 = 120 cm²; total area = 120 + 16√3 cm².