Lesson 74 of 100 | Quantitative Aptitude / Solid Mensuration / ठोस क्षेत्रमिति
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
- A right prism has a right-triangular base with sides 5, 12 and 13 cm, and height 10 cm. Find total area and volume.
- A right prism has square base side 7 cm and height 12 cm. Find total area and volume.
- A right square pyramid has base side 12 cm and perpendicular height 8 cm. Find total area and volume.
- A regular triangular pyramid has equilateral base side 8 cm and face slant height 10 cm. Find lateral and total areas.
Worked solutions
- B = 5 × 12 ÷ 2 = 30 cm²; p = 30 cm. Total area = 30 × 10 + 60 = 360 cm²; volume = 30 × 10 = 300 cm³.
- B = 49 cm²; p = 28 cm. Total area = 28 × 12 + 98 = 434 cm²; volume = 49 × 12 = 588 cm³.
- s = √(8² + 6²) = 10 cm. Total area = 48 × 10 ÷ 2 + 144 = 384 cm²; volume = 144 × 8 ÷ 3 = 384 cm³.
- 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.