The volume of a solid makes sense if you picture stacking coins.
Stack coins of area πr² to a height h and you get a cylinder with volume πr²h; a cone with the same base and height holds a third of that.
A sphere that fits snugly inside a cylinder holds two thirds of it, so when the height equals the diameter, the volumes of cylinder, cone and sphere are in the ratio 3 : 1 : 2.
On this page you change the radius and height and switch between cylinder, cone and sphere to compare their volumes with π left in, and unfold surfaces to see where the area formulas come from.
Solids built from stacked coins
seegongsik.com
Cylinder, cone or sphere drawn to the same scale, with its volume
3
6
🪙 Stacking coins
①A coin of radius r has area πr²
②Stack such coins to a height h and you get a cylinder, so its volume is πr² × h
③A cone is like a stack of coins that shrink towards the top, so it holds less than a cylinder with the same base and height: exactly a third
④The picture draws radius and height to the same scale. With r = 3 and h = 6, the cylinder is as tall as it is wide
Volume and surface area of prisms and cylinders
Volume of a prism or cylinder
V = Sh = (base area) × (height)
Whatever the shape of the base, multiply the base area by the height
Volume of a cylinder
V = πr²h
A cylinder whose base is a circle of radius r
Surface area of a cylinder
S = 2πr² + 2πrh = 2πr(r + h)
The two circular ends, plus the curved side, which unrolls into a rectangle 2πr wide and h tall
🔍 Unrolling the side
①Cut the curved side of a cylinder straight down and unroll it: it is a rectangle
②Its width is the circumference of the base, 2πr
③Its height is h
④So the curved area is 2πrh, and adding the two ends, 2πr², gives the total surface area
Why a cone is one third
Volume of a pyramid or cone
V = 13Sh
One third of the prism or cylinder with the same base and height
Volume of a cone
V = 13πr²h
One third of the cylinder's volume πr²h
Curved surface of a cone
S = πrl
l is the slant height. Unrolled, the curved surface is a sector with radius l and arc 2πr, so its area is (1/2) × l × 2πr = πrl
💡 Checking by experiment
①Take a cylinder-shaped container and a cone-shaped one with the same base and height
②Fill the cone with water and pour it into the cylinder: it takes three pourings to fill it
③You can see with your own eyes, without calculating, that the cone holds a third of the cylinder
④Choose the cone in the picture and its volume comes out as 1/3 of the cylinder's
Volume and surface area of a sphere
seegongsik.com
Comparing the volumes of a cylinder, cone and sphere with the same r and h
Volume of a sphere
V = 43πr³
A sphere of radius r is 2/3 of a cylinder of height 2r
Surface area of a sphere
S = 4πr²
Equal to the area of four circles of the same radius
🏀 A sphere is two thirds of its cylinder
①Put a sphere into the cylinder it fits exactly, whose height is the diameter 2r
②That cylinder's volume is πr² × 2r = 2πr³
③The sphere's volume is (4/3)πr³, which is 2/3 of the cylinder
④The cone with the same base and height is (1/3) × 2πr³ = (2/3)πr³, so cylinder : cone : sphere = 3 : 1 : 2, a relationship Archimedes discovered
Check with examples
Example 1
A cylinder has base radius 3 and height 5. Find its volume. (Leave π in the answer.)
1
The volume of a cylinder is (base area) × (height) = πr² × h.
volume = π × 3² × 5
2
Calculate.
= 9π × 5 = 45π
▸ 45π
Set r = 3 and h = 5 in the picture and choose the cylinder: its volume shows 45π ≈ 141.4.
Example 2
A square-based pyramid has a base of side 6 and height 10. Find its volume.
1
The volume of a pyramid is (1/3) × (base area) × (height), and the base area is 6² = 36.
volume = (1/3) × 36 × 10
2
Calculate.
= (1/3) × 360 = 120
▸ 120
A square prism with the same base and height has volume 360, and the pyramid is a third of it, 120.