Sec 2 · G2 (N(A))

Pyramids, cones and spheres

Practise pyramids, cones and spheres with step-by-step help. Free account.

1. Formulae

A pyramid with its perpendicular height h; a cone with radius r, height h and slant height l; a sphere with radius r.

Solid	Volume	Surface area
pyramid	$\dfrac{1}{3} \times \text{base area} \times h$	base $+$ the triangular faces
cone	$\dfrac{1}{3}\pi r^{2}h$	curved: $\pi rl$ ; total: $\pi rl + \pi r^{2}$
sphere	$\dfrac{4}{3}\pi r^{3}$	$4\pi r^{2}$
hemisphere	$\dfrac{2}{3}\pi r^{3}$	curved: $2\pi r^{2}$ ; solid (with the flat circle): $3\pi r^{2}$

2. Worked examples

Square pyramid: base 6 cm, height 10 cm

\displaystyle V = \dfrac{1}{3} \times 6^{2} \times 10

e.g.

\displaystyle \dfrac{1}{3} \times 6^{2} \times 10

\displaystyle = 120

$V = 120$ cm³

Cone: r = 3, h = 4, so the slant height l = 5

\displaystyle \pi rl + \pi r^{2}

e.g.

\displaystyle \pi(3)(5) + \pi(3)^{2}

\displaystyle = 24\pi

\displaystyle = 75.4\text{ (3 s.f.)}

Sphere, radius 6 cm

\displaystyle V = \dfrac{4}{3}\pi r^{3}

e.g.

\displaystyle \dfrac{4}{3}\pi(6)^{3}

\displaystyle = 288\pi

\displaystyle = 905\text{ (3 s.f.)}

$V = 905$ cm³ (3 s.f.)

📖 Worked example

The diagram shows a conical party hat of height $8$ cm and base diameter $7$ cm. Find its volume, correct to 3 significant figures.

Step 1 of 2

The radius is half the diameter

r = \dfrac{7}{2}

= 3.5

📖 Worked example

The diagram shows a solid sphere of diameter $7$ cm. Find its volume, correct to 3 significant figures.

Step 1 of 2

The radius is half the diameter

r = \dfrac{7}{2}

= 3.5

📖 Worked example

The diagram shows a solid made of a cone on top of a hemisphere. The cone has height $10$ cm and the common radius is $2$ cm. Find the volume of the solid, correct to 3 significant figures.

Step 1 of 3

Volume of the cone = ⅓πr²h

Volume of cone

= \dfrac{1}{3} \times \pi \times 2^2 \times 10

= 41.8879\ldots

cm³

Practise this →

3. Containers, melting and frustums

📖 Worked example

A cylindrical container of radius $5$ cm contains some water. A solid metal ball of radius $2$ cm is placed in the water and sinks to the bottom, completely covered by the water, with none spilling out. Find the rise in the water level, correct to 3 significant figures.

Step 1 of 2

The water rises by the volume of the ball: volume of the ball

Volume of the ball

= \dfrac{4}{3} \times \pi \times 2^3

= 33.5103\ldots

cm³

📖 Worked example

The diagram shows a lampshade stand in the shape of a frustum: a large cone with a small cone cut off its top. The circular faces have radii $4$ cm and $2$ cm, and the frustum is $5$ cm high. Find the volume of the frustum, correct to 3 significant figures.

Step 1 of 4

Similar triangles: the small cone's height over its radius is the large cone's height over its radius

\dfrac{h}{2} = \dfrac{h + 5}{4}

4h = 2h + 10

h = 5

cm, so the large cone is

10

cm high.

Practise this →

Practise pyramids, cones and spheres with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free

1. Formulae

Solid	Volume	Surface area
pyramid	$\dfrac{1}{3} \times \text{base area} \times h$	base $+$ the triangular faces
cone	$\dfrac{1}{3}\pi r^{2}h$	curved: $\pi rl$ ; total: $\pi rl + \pi r^{2}$
sphere	$\dfrac{4}{3}\pi r^{3}$	$4\pi r^{2}$
hemisphere	$\dfrac{2}{3}\pi r^{3}$	curved: $2\pi r^{2}$ ; solid (with the flat circle): $3\pi r^{2}$

2. Worked examples

Square pyramid: base 6 cm, height 10 cm

\displaystyle V = \dfrac{1}{3} \times 6^{2} \times 10

e.g.

\displaystyle \dfrac{1}{3} \times 6^{2} \times 10

\displaystyle = 120

$V = 120$ cm³

Cone: r = 3, h = 4, so the slant height l = 5

\displaystyle \pi rl + \pi r^{2}

e.g.

\displaystyle \pi(3)(5) + \pi(3)^{2}

\displaystyle = 24\pi

\displaystyle = 75.4\text{ (3 s.f.)}

Sphere, radius 6 cm

\displaystyle V = \dfrac{4}{3}\pi r^{3}

e.g.

\displaystyle \dfrac{4}{3}\pi(6)^{3}

\displaystyle = 288\pi

\displaystyle = 905\text{ (3 s.f.)}

$V = 905$ cm³ (3 s.f.)

📖 Worked example

The diagram shows a conical party hat of height $8$ cm and base diameter $7$ cm. Find its volume, correct to 3 significant figures.

Step 1 of 2

The radius is half the diameter

r = \dfrac{7}{2}

= 3.5

📖 Worked example

The diagram shows a solid sphere of diameter $7$ cm. Find its volume, correct to 3 significant figures.

Step 1 of 2

The radius is half the diameter

r = \dfrac{7}{2}

= 3.5

📖 Worked example

The diagram shows a solid made of a cone on top of a hemisphere. The cone has height $10$ cm and the common radius is $2$ cm. Find the volume of the solid, correct to 3 significant figures.

Step 1 of 3

Volume of the cone = ⅓πr²h

Volume of cone

= \dfrac{1}{3} \times \pi \times 2^2 \times 10

= 41.8879\ldots

cm³

Practise this →

3. Containers, melting and frustums

📖 Worked example

Step 1 of 2

The water rises by the volume of the ball: volume of the ball

Volume of the ball

= \dfrac{4}{3} \times \pi \times 2^3

= 33.5103\ldots

cm³

📖 Worked example

Step 1 of 4

Similar triangles: the small cone's height over its radius is the large cone's height over its radius

\dfrac{h}{2} = \dfrac{h + 5}{4}

4h = 2h + 10

h = 5

cm, so the large cone is

10

cm high.

Practise this →