Volume of solids
Volume measures the space a solid occupies, in cubic units. Prisms and cylinders, pyramids and cones, and spheres follow three related formulas.
Prisms and cylinders
A prism or cylinder has volume base area times height, \(V = Bh\). A box 5 by 4 by 3 holds 60 cubic units; a cylinder of radius 4 and height 10 holds \(16\pi \times 10 = 160\pi \approx 503\) cubic units. Volume is measured in cubic units, and 1 litre is 1,000 cubic centimetres.
Cavalieri's principle
Bonaventura Cavalieri stated in 1635 that solids with the same height and equal cross-sectional areas at every level have equal volumes. A straight stack of coins and a slanted stack of the same coins have the same volume, which is why oblique prisms and cylinders also use V = Bh.
Pyramids and cones
A pyramid or cone has one third the volume of the prism or cylinder with the same base and height: \(V = \tfrac{1}{3}Bh\). A cone of radius 3 and height 4 holds \(\tfrac{1}{3} \times 9\pi \times 4 = 12\pi \approx 37.7\) cubic units. Three cones of water exactly fill the matching cylinder.
Spheres
A sphere's volume is \(V = \tfrac{4}{3}\pi r^3\). A ball of radius 3 cm holds \(36\pi \approx 113\) cubic centimetres. Archimedes showed that a sphere has two thirds the volume of the cylinder that just contains it.
Composite solids
Add or subtract volumes of simpler parts. A grain silo made of a cylinder of radius 3 m and height 10 m topped by a hemisphere holds 90 pi + (2/3) pi (27) = 108 pi, about 339 cubic metres.
Scaling volume
Multiplying every length by k multiplies volume by k cubed. A ball with twice the radius holds 8 times as much. This is why a 30 cm cake needs far more batter than a 20 cm cake of the same depth: area, and so batter, scales by (30/20) squared = 2.25.
Density and capacity
Density is mass per unit volume. Water has a density of about 1 gram per cubic centimetre, so a 1-litre bottle of water has a mass of about 1 kilogram. Knowing a container's volume gives its capacity and the mass of what fills it.
Worked example
A swimming pool is a rectangular prism 25 m long, 10 m wide, and 2 m deep. Its volume is 500 cubic metres, which is 500,000 litres, since 1 cubic metre holds 1,000 litres. The water's mass is about 500,000 kg, or 500 tonnes. Filling it with a hose delivering 20 litres per minute would take about 25,000 minutes, roughly 17 days.
Key takeaways
Volume is measured in cubic units, and 1 litre is 1,000 cubic centimetres. Prisms and cylinders have volume base area times height, also for oblique solids by Cavalieri's principle (1635). Pyramids and cones have one third of that. Spheres have volume four thirds pi r cubed, two thirds of their enclosing cylinder, as Archimedes showed. Composite solids add or subtract parts. Scaling lengths by k scales volume by k cubed. Water's density of about 1 g per cubic centimetre links volume to mass.
Flashcard
Base area x height
Check your understanding
What is the volume of a cone with radius 3 and height 4?