Volume of Solids
Senior 2 · Mathematics
Volume of Solids
Lesson 3 of 4
22 min
A heap of sand at a building site, the roof of a round traditional house, a funnel, the Great Pyramid of Giza: all of these come to a point. Solids that rise from a base to a single point, the apex, hold exactly one third as much as the prism or cylinder with the same base and height. This lesson shows that fact and how to use it.
A pyramid has a polygon base and triangular faces that meet at the apex. A cone is the same idea with a circular base. The height h is the perpendicular distance from the apex to the base, not the length of a sloping edge.
A cube can be cut into three identical pyramids, each with one face of the cube as its base and the opposite corner as its apex. Each pyramid therefore has one third of the cube's volume. The same is true for every pyramid compared with the prism that has the same base and height, and for every cone compared with the matching cylinder:
Pyramid: V = ⅓ × base area × h
Cone: V = ⅓ πr2h
1. A square-based pyramid has base edge 6 m and height 4 m. Base area = 36 m2, so V = ⅓ × 36 × 4 = 48 m3.
2. A cone has radius 3 cm and height 10 cm. V = ⅓ × 3.14 × 9 × 10 = 94.2 cm3. The cylinder with the same base and height would hold 282.6 cm3, three times as much.
3. A heap of sand. At a building site in Musanze, sand is tipped into a cone-shaped heap 4 m across and 1.5 m high. The radius is 2 m, so V = ⅓ × 3.14 × 4 × 1.5 = 6.28 m3.
4. Finding a height. A cone of radius 4 cm has volume 100.48 cm3. Then ⅓ × 3.14 × 16 × h = 100.48, so 16.7467h = 100.48 and h = 6 cm.
Tip
Memory hook. "Pointed solids take a third": pyramid and cone = ⅓ of the matching prism or cylinder. Use the vertical height, never the slant height. If a question gives only the slant height, find the vertical height first with Pythagoras' theorem (Unit 6).
A pyramid has a rectangular base 5 cm by 3 cm and a height of 8 cm. Find its volume.
Solution: ⅓ × 15 × 8 = 40 cm3.
A cone has radius 6 cm and height 8 cm. Find its volume (π = 3.14).
Solution: ⅓ × 3.14 × 36 × 8 = 301.44 cm3.
A square pyramid has base edge 10 m and height 9 m. Find its volume.
Solution: ⅓ × 100 × 9 = 300 m3.
A cylinder and a cone have the same radius and the same height. The cylinder holds 90 cm3. How much does the cone hold?
Solution: One third: 30 cm3.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
