What is a solid of revolution?
Take a flat shape. Put a straight line next to it. This line is the axis. Now turn the shape one full turn (360°) around the axis. Every point of the shape draws a circle. All these circles together make a solid. We call it a solid of revolution (revolution = one full turn).
- Rectangle turned about one side → cylinder.
- Right triangle turned about one leg → cone.
- Half-circle turned about its diameter → sphere.
- Right trapezium turned about its straight side → frustum.
Any cut through the axis (an axial section) shows the original shape twice, mirror to mirror. A cut at right angles to the axis is always a circle.
Cylinder and cone
A cylinder has two equal circle ends, radius r, and height h.
- Volume V = πr²h (area of the base × height).
- Curved surface area = 2πrh. Total surface area = 2πr(r + h).
A cone has one circle base and a tip. Its slant height is l = √(r² + h²) (Pythagoras on the triangle).
- Volume V = ⅓πr²h.
- Curved surface area = πrl. Total = πr(l + r).
Why one third?
Fill a cone with water and pour it into a cylinder with the same r and h. It takes three cones to fill the cylinder.
Sphere, ball and frustum
A sphere is the surface; a ball is the solid inside it. For radius r:
- Volume V = ⁴⁄₃πr³.
- Surface area A = 4πr² (four times the area of its biggest circle).
- A hemisphere (half ball) has V = ⅔πr³ and total surface area 3πr².
A frustum is what is left when you cut the top off a cone with a cut parallel to the base. Big radius R, small radius r, height h, slant l = √(h² + (R − r)²):
- V = ⅓πh(R² + Rr + r²).
- Curved surface area = π(R + r)l.
Combinations of solids
Many real objects join two or more round solids: a capsule (cylinder + two hemispheres), an ice-cream (cone + hemisphere), a tent (cylinder + cone).
- Split the object into simple solids.
- Find each volume and add (or subtract if one is cut out).
- For surface area, add only the outside surfaces you can touch. Do not count the joined faces.
Volume by calculus: the disc method
Turn the curve y = f(x), from x = a to x = b, around the x-axis. Slice the solid into thin discs. One disc has radius y and thickness dx, so its volume is πy²dx. Add all discs:
V = π ∫ₐᵇ y² dx
Turning around the y-axis instead: V = π ∫ x² dy. If the region has a hole (between two curves, outer y₁ and inner y₂) use washers: V = π ∫ (y₁² − y₂²) dx.
Check: y = r (a flat line) from 0 to h gives π r² h, the cylinder. y = (r/h)x gives ⅓πr²h, the cone.
Mean value of f on [a, b] = (1/(b − a)) ∫ₐᵇ f(x) dx, the height of the rectangle with the same area.
Key formulas and definitions
- Cylinder: V = πr²h, CSA = 2πrh
- Cone: V = ⅓πr²h, l = √(r² + h²), CSA = πrl
- Sphere: V = ⁴⁄₃πr³, A = 4πr²
- Frustum: V = ⅓πh(R² + Rr + r²), CSA = π(R + r)l
- Disc method: V = π∫ y² dx
- Washer method: V = π∫ (y₁² − y₂²) dx
Worked examples
1. A rectangle 3 cm by 5 cm is turned about its 5 cm side. Find the volume.
The 5 cm side is the axis, so h = 5 and r = 3. V = πr²h = π × 9 × 5 = 45π ≈ 141.4 cm³.
2. A right triangle with legs 6 cm and 8 cm is turned about the 8 cm leg. Find the volume and curved surface area.
Cone: r = 6, h = 8. V = ⅓π × 36 × 8 = 96π ≈ 301.6 cm³. Slant l = √(36 + 64) = 10. CSA = π × 6 × 10 = 60π ≈ 188.5 cm².
3. A half-circle of radius 3 cm turns about its diameter. Find the volume and surface area.
Sphere with r = 3. V = ⁴⁄₃π × 27 = 36π ≈ 113.1 cm³. A = 4π × 9 = 36π ≈ 113.1 cm².
4. A bucket is a frustum with radii 15 cm and 10 cm and height 24 cm. How many litres does it hold?
V = ⅓π × 24 × (225 + 150 + 100) = 8π × 475 = 3800π ≈ 11 938 cm³ ≈ 11.9 litres (1 litre = 1000 cm³).
5. A capsule is a cylinder 10 mm long with a hemisphere of radius 3 mm on each end. Find its volume.
Two hemispheres = one sphere: ⁴⁄₃π × 27 = 36π. Cylinder: π × 9 × 10 = 90π. Total = 126π ≈ 395.8 mm³.
6. Find the volume when y = √x, 0 ≤ x ≤ 4, is turned about the x-axis.
V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = π × 8 = 8π ≈ 25.1 cubic units.
Common mistakes
- Using the slant height l instead of the vertical height h in V = ⅓πr²h. Volume always uses h.
- Mixing up which side is the axis. The side on the axis becomes the height; the other side becomes the radius.
- Forgetting to square y in the disc method: it is π∫y² dx, not π∫y dx.
- Counting the joined faces when finding the surface area of a combined solid.