Building on cuboids and cylinders
You already know the cuboid (volume lbh) and the cylinder (volume πr²h, curved surface area 2πrh). This chapter adds two curved solids: the cone and the sphere.
The right circular cone
A right circular cone has a circular base of radius r, a vertex directly above the centre of the base at height h, and a curved surface joining them. An ice-cream cone and a birthday cap are cones.
The distance from the vertex to any point on the edge of the base is the slant height l. The height, radius and slant height form a right triangle, so by Pythagoras:
l = √(h² + r²)
Formula | |
|---|---|
Curved surface area | πrl |
Total surface area | πr(l + r) (curved surface plus the base) |
Volume | ⅓πr²h |
A cone holds exactly one-third of a cylinder with the same base and height. Fill such a cone with sand and pour it into the cylinder: it takes three fillings.
Example. A cone has radius 7 cm and height 24 cm (π = 22/7).
l = √(24² + 7²) = √(576 + 49) = √625 = 25 cm
Curved surface area = 22/7 × 7 × 25 = 550 cm²
Total surface area = 22/7 × 7 × (25 + 7) = 704 cm²
Volume = ⅓ × 22/7 × 7 × 7 × 24 = 1232 cm³
Example: a tent. A conical tent is 10 m high with a base radius of 24 m. Its slant height is √(100 + 576) = 26 m, and the canvas needed for the curved surface is 22/7 × 24 × 26 ≈ 1961 m².
The sphere
A sphere is the set of all points in space at the same distance r (the radius) from a centre. A ball is a sphere.
Formula | |
|---|---|
Surface area of a sphere | 4πr² |
Volume of a sphere | ⁴⁄₃πr³ |
A neat way to remember the surface area: the curved surface of a sphere equals four circles of the same radius. Wind a string over a hemisphere and then fill circles of the same radius with it to see this.
Example. A ball of radius 7 cm:
Surface area = 4 × 22/7 × 7 × 7 = 616 cm²
Volume = ⁴⁄₃ × 22/7 × 7 × 7 × 7 = 4312/3 ≈ 1437.3 cm³
The hemisphere
Cutting a sphere through its centre gives two hemispheres. A bowl is a hemisphere.
Formula | |
|---|---|
Curved surface area | 2πr² |
Total surface area (solid, with flat face) | 3πr² |
Volume | ⅔πr³ |
Example. A hemispherical bowl of inner radius 3.5 cm holds ⅔ × 22/7 × 3.5³ ≈ 89.8 cm³, which is about 89.8 mL of soup.
How changes in radius affect a sphere
Surface area depends on r², and volume on r³. So if the radius is doubled:
surface area becomes 2² = 4 times as large,
volume becomes 2³ = 8 times as large.
If a balloon's radius grows from 7 cm to 14 cm, the ratio of its surface areas is 1 : 4.
Common mistakes
Using height h instead of slant height l in πrl.
Forgetting the ⅓ in the cone's volume or the ⁴⁄₃ in the sphere's.
For a hollow hemisphere (a bowl), the total surface area is just 2πr²; the 3πr² formula is for a solid hemisphere.
Using diameter for radius.
Summary
Cone: l = √(h² + r²); CSA = πrl; TSA = πr(l + r); V = ⅓πr²h.
Sphere: SA = 4πr²; V = ⁴⁄₃πr³.
Hemisphere: CSA = 2πr²; TSA = 3πr²; V = ⅔πr³.