Volume of Solids
Senior 2 · Mathematics
Volume of Solids
Lesson 4 of 4
24 min
Footballs, oranges, water droplets and planets are close to perfect spheres. Grain silos and many water towers are cylinders topped with a half-sphere. To find how much such objects hold, you need the volume of a sphere and the skill of splitting a composite solid into simple parts.
A sphere is the set of all points in space at the same distance r (the radius) from a centre. Cut it through the centre and you get two hemispheres.
Picture a sphere that fits exactly inside a cylinder: the cylinder has radius r and height 2r.
The cylinder holds πr2 × 2r = 2πr3. A cone with the same base and height would hold one third of that, ⅔πr3. Archimedes proved that the sphere fills exactly two thirds of this cylinder. Equivalently, the sphere holds what the cylinder holds minus what the cone holds: 2πr3 − ⅔πr3. So
Sphere: V = (4 ÷ 3)πr3 and hemisphere: V = (2 ÷ 3)πr3.
In other words, the sphere fills two thirds of the cylinder around it.
1. A football has radius 11 cm. V = (4 ÷ 3) × 3.14 × 113 = (4 ÷ 3) × 3.14 × 1331 ≈ 5,572.45 cm3, about 5.6 litres of air.
2. A hemispherical bowl has radius 9 cm. V = (2 ÷ 3) × 3.14 × 729 = 1,526.04 cm3, about 1.5 litres.
3. Finding the radius. A sphere has volume 36π cm3. Then (4 ÷ 3)πr3 = 36π, so r3 = 27 and r = 3 cm.
Split the solid into parts whose formulas you know, find each volume, then add (or subtract for a hole).
Worked example 4
a grain silo. A maize silo in Kayonza is a cylinder of radius 3 m and height 10 m with a hemispherical roof.
Worked example 5
an ice-cream cone. A cone of radius 3 cm and height 10 cm is topped with a hemisphere of ice cream of radius 3 cm. Total = 94.2 + 56.52 = 150.72 cm3.
Tip
Exam technique. Cube the radius in the sphere formula (r3, not r2); squaring it gives an area, not a volume. For a composite solid, write one line per part with a label ("cylinder: …", "hemisphere: …") and add at the end; you earn method marks for each part even if one calculation slips.
Find the volume of a sphere of radius 10 cm (π = 3.14).
Solution: (4 ÷ 3) × 3.14 × 1000 ≈ 4,186.67 cm3.
Find the volume of a sphere of radius 6 cm in terms of π.
Solution: (4 ÷ 3)π × 216 = 288π cm3.
Show that a cylinder of radius 3 cm and height 4 cm has the same volume as a sphere of radius 3 cm.
Solution: Cylinder: π × 9 × 4 = 36π. Sphere: (4 ÷ 3)π × 27 = 36π. They are equal.
A solid is a hemisphere of radius 3 cm on top of a cylinder of radius 3 cm and height 5 cm. Find its volume in terms of π.
Solution: Cylinder: 45π. Hemisphere: (2 ÷ 3)π × 27 = 18π. Total: 63π cm3.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
