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Cones and spheres — 0580

Keep perpendicular cone height separate from its slant height.

Core & Extended · Syllabus C5.4, E5.4

Notes and key methods

A cone has volume \(\displaystyle \frac{\pi r^{2}h}{3}\) and curved surface area πrl, where h is perpendicular height and l is slant height. For a right circular cone, \(\displaystyle l^{2}=r^{2}+h^{2}\). Total surface area also includes its circular base.

A sphere has volume \(\displaystyle \frac{4\pi r^3}{3}\) and surface area \(\displaystyle 4\pi r^{2}\). These solids belong to Core as well as Extended. State whether a requested surface includes a base or cut face before applying a formula.

Check the conditions

Three worked examples

Example 1 · Cones and spheres

A right cone has radius 18 cm and perpendicular height 24 cm. Separately, a sphere has radius 3 cm.

  1. Find the cone slant height in cm.
  2. Write the cone volume as \(\displaystyle k\pi \mathrm{cm}^{3}\). Find k.
  3. Write the cone total surface area as \(\displaystyle k\pi \mathrm{cm}^{2}\). Find k.
  4. Write the sphere volume as \(\displaystyle k\pi \mathrm{cm}^{3}\). Find k.
Show the reasoning

(a)

Use the right triangle between radius, perpendicular height and slant height.

Pythagoras gives \(\displaystyle \sqrt{r^{2}+h^{2}}=30 \mathrm{cm}\).

Answer: \(30\).

(b)

Use perpendicular height in the cone volume formula.

The coefficient \(\displaystyle \frac{r^{2}h}{3}\) is \(2592\).

Answer: \(2592\).

(c)

Include both the slant-height curved surface and circular base.

Curved area plus base gives coefficient rl+\(\displaystyle r^{2}\)=864.

Answer: \(864\).

(d)

Cube the sphere radius and retain the factor four thirds.

The coefficient \(\displaystyle \frac{4R^{3}}{3}\) is \(36\).

Answer: \(36\).

Example 2 · Cones and spheres

A right cone has radius 3 cm and perpendicular height 4 cm. Separately, a sphere has radius 4 cm.

  1. Find the cone slant height in cm.
  2. Write the cone volume as \(\displaystyle k\pi \mathrm{cm}^{3}\). Find k.
  3. Write the cone total surface area as \(\displaystyle k\pi \mathrm{cm}^{2}\). Find k.
  4. Write the sphere volume as \(\displaystyle k\pi \mathrm{cm}^{3}\). Find k.
Show the reasoning

(a)

Use the right triangle between radius, perpendicular height and slant height.

Pythagoras gives \(\displaystyle \sqrt{r^{2}+h^{2}}=5 \mathrm{cm}\).

Answer: \(5\).

(b)

Use perpendicular height in the cone volume formula.

The coefficient \(\displaystyle \frac{r^{2}h}{3}\) is \(12\).

Answer: \(12\).

(c)

Include both the slant-height curved surface and circular base.

Curved area plus base gives coefficient rl+\(\displaystyle r^{2}\)=24.

Answer: \(24\).

(d)

Cube the sphere radius and retain the factor four thirds.

The coefficient \(\displaystyle \frac{4R^{3}}{3}\) is \(\displaystyle \frac{256}{3}\).

Answer: \(\displaystyle \frac{256}{3}\).

Example 3 · Cones and spheres

A right cone has radius 6 cm and perpendicular height 8 cm. Separately, a sphere has radius 5 cm.

  1. Find the cone slant height in cm.
  2. Write the cone volume as \(\displaystyle k\pi \mathrm{cm}^{3}\). Find k.
  3. Write the cone total surface area as \(\displaystyle k\pi \mathrm{cm}^{2}\). Find k.
  4. Write the sphere volume as \(\displaystyle k\pi \mathrm{cm}^{3}\). Find k.
Show the reasoning

(a)

Use the right triangle between radius, perpendicular height and slant height.

Pythagoras gives \(\displaystyle \sqrt{r^{2}+h^{2}}=10 \mathrm{cm}\).

Answer: \(10\).

(b)

Use perpendicular height in the cone volume formula.

The coefficient \(\displaystyle \frac{r^{2}h}{3}\) is \(96\).

Answer: \(96\).

(c)

Include both the slant-height curved surface and circular base.

Curved area plus base gives coefficient rl+\(\displaystyle r^{2}\)=96.

Answer: \(96\).

(d)

Cube the sphere radius and retain the factor four thirds.

The coefficient \(\displaystyle \frac{4R^{3}}{3}\) is \(\displaystyle \frac{500}{3}\).

Answer: \(\displaystyle \frac{500}{3}\).

Common mistakes

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