Solids of Revolution: Cylinders, Cones and Spheres
Solids of revolution are formed by rotating a plane figure about a line: a rectangle gives a cylinder, a right triangle gives a cone and a semicircle gives a sphere. In brief: with radius r and height h, a cylinder has volume πr2h, a cone has volume (1/3)πr2h and a sphere has volume (4/3)πr3. The surface areas are 2πr(r + h) for a closed cylinder, πr(r + s) for a cone with slant height s and 4πr2 for a sphere.
These curved solids model cans, funnels, balls, tanks and planets. Their formulas contain π and are related: a cone and a hemisphere together fill a cylinder when the height equals the radius, as noticed by Archimedes. This guide describes the three solids, their formulas, worked examples, the relationship between them and the idea of rotating a figure, with a table of formulas.
What are the formulas for a cylinder?
A right circular cylinder has two parallel circular bases of radius r and a height h. Rolled out, the lateral surface is a rectangle with sides 2πr (the circumference) and h.
- Volume V = πr2h (base area times height)
- Lateral area L = 2πrh
- Total surface area S = 2πr2 + 2πrh = 2πr(r + h)
What are the formulas for a cone?
A right circular cone has a circular base of radius r, a height h (from the apex perpendicular to the base) and a slant height s = √(r2 + h2). Rolled out, the lateral surface is a sector of a circle with radius s and arc length 2πr.
- Volume V = (1/3)πr2h
- Lateral area L = πrs
- Total surface area S = πr2 + πrs = πr(r + s)
The central angle of the unrolled sector is θ = 2πr/s in radians. For r = 3 and s = 5 it is 6π/5 = 216°.
What are the formulas for a sphere?
A sphere of radius r is the set of points at distance r from the center.
- Volume V = (4/3)πr3
- Surface area S = 4πr2, which is four times the area of a great circle.
- Hemisphere: V = (2/3)πr3, curved surface 2πr2, total surface 3πr2.
If the radius doubles, the surface area is multiplied by 4 and the volume by 8.
How are the three solids related?
Take a sphere of radius r, a cylinder of radius r and height 2r that just contains it and a cone with radius r and height 2r. Their volumes are (4/3)πr3, 2πr3 and (2/3)πr3. The ratio cone : sphere : cylinder is 1 : 2 : 3. Also the surface area of the sphere, 4πr2, equals the lateral area of the enclosing cylinder, 2πr · 2r = 4πr2. Archimedes was proud of this discovery and is said to have wished for the figure of the sphere in a cylinder on his tomb.
Worked examples
Example 1: a can
A can has radius 4 cm and height 12 cm. V = π · 16 · 12 = 192π ≈ 603.2 cm3 (about 0.6 litres). Lateral area: 2π · 4 · 12 = 96π ≈ 301.6 cm2. Total area: 96π + 32π = 128π ≈ 402.1 cm2.
Example 2: a cone
A cone has radius 6 and height 8. The slant height is s = 10. V = (1/3)π · 36 · 8 = 96π ≈ 301.6. L = π · 6 · 10 = 60π ≈ 188.5. S = 60π + 36π = 96π ≈ 301.6. The numerical values of V and S coincide, but they have different units.
Example 3: a sphere
A ball has a diameter of 22 cm, so r = 11. V = (4/3)π · 1331 ≈ 5,575.3 cm3 and S = 4π · 121 ≈ 1,520.5 cm2.
Example 4: finding the radius from the volume
A sphere has volume 36π. Then (4/3)r3 = 36, so r3 = 27 and r = 3. The surface area is 4π · 9 = 36π. (Again the numerical values coincide.)
Example 5: melting
A metal sphere of radius 6 cm is melted into small spheres of radius 1 cm. The number of small spheres is (6/1)3 = 216, since volume scales with the cube of the radius. The total surface area of the small spheres is 216 · 4π = 864π, which is 6 times the original 144π.
Example 6: a cylinder with a given volume and minimal area
For a closed can with volume 500 cm3, the surface area S = 2πr2 + 1000/r is smallest when r = ∛(250/π) ≈ 4.30 cm, where h = 2r ≈ 8.60 cm. A can with height equal to its diameter uses the least material. This is found with calculus.
Formula table
| Solid | Volume | Curved (lateral) area | Total surface area |
|---|---|---|---|
| Cylinder (r, h) | πr2h | 2πrh | 2πr(r + h) |
| Cone (r, h, s) | (1/3)πr2h | πrs | πr(r + s) |
| Sphere (r) | (4/3)πr3 | 4πr2 | 4πr2 |
| Hemisphere (r) | (2/3)πr3 | 2πr2 | 3πr2 |
Where do these results come from?
Archimedes, in the third century BCE, proved the formulas for the volume and surface area of the sphere in his work On the Sphere and Cylinder, using the method of exhaustion. Eudoxus had earlier proved the volume of the cone as one third of the cylinder. Integral calculus later gave a unified method: the volume of a solid of revolution is V = π ∫ f(x)2 dx.
Where are they used today?
Cylinders are tanks, pipes and cans. A water tank with radius 1.5 m and height 3 m holds π · 2.25 · 3 ≈ 21.2 m3, or 21,200 litres. Cones appear as funnels, traffic cones and silos, and spheres as balls, bubbles and planets. The Earth has a radius of about 6,371 km, so its surface area is about 4π · 63712 ≈ 510 million km2.
Which mistakes are common?
Using the diameter in place of the radius
All formulas use the radius. Halve the diameter first.
Using the height instead of the slant height
The lateral area of a cone is πrs, with s = √(r2 + h2), not πrh.
Forgetting the factor 1/3 for a cone
A cone has one third of the volume of the cylinder with the same base and height.
Mixing up volume and surface area units
Volume is in cubic units and area is in square units. Compare only quantities of the same kind.