Sec 2 · G3 (Express)

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.

📖 Worked example

The diagram shows a solid pyramid with a square base of side $6$ cm. Its apex $V$ is vertically above the centre $O$ of the base, and each of its four slant edges is $7$ cm long. Find the height of the pyramid, and hence its volume, correct to 3 significant figures.

Step 1 of 3

Half the diagonal of the base: O is the centre, so OB is half of BD (Pythagoras on the base)

OB^2 = 3^2 + 3^2

= 18

📖 Worked example

The diagram shows a sector of a circle of radius $6$ cm and angle $120^\circ$ ( $\dfrac{1}{3}$ of the circle). Its two straight edges are joined, without overlapping, to make a paper cup. Find the height of the cone, correct to 3 significant figures.

Step 1 of 2

The arc of the sector becomes the circumference of the cone's base

\dfrac{120}{360} \times 2\pi \times 6 = 2\pi r

r = 2

📖 Worked example

A solid sphere of radius $r$ cm has the same volume as a solid cone of base radius $\dfrac{3}{2}r$ cm and height $h$ cm. Express $h$ in terms of $r$ .

Step 1 of 3

Volume of the sphere: ⁴⁄₃πr³

Volume

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

📖 Worked example

A solid metal cylinder of radius $6$ cm and height $10$ cm is melted down and recast into small solid spheres, each of radius $0.8$ cm. Find the greatest number of spheres that can be made.

Step 1 of 3

The volume of metal (melting doesn't change it)

Volume of the cylinder

= \pi \times 6^2 \times 10

= 1130.9733\ldots

cm³

📖 Worked example

The curved surface of a hemispherical dome of radius $6$ m is to be painted. One tin of paint covers $8$ m². Find the number of tins of paint that must be bought.

Step 1 of 3

Area of the curved surface of the dome: 2πr²

Area

= 2 \times \pi \times 6^2

= 226.1946\ldots

m²

Solid	Surface to count
half a cone (cut through its axis)	$\dfrac{1}{2}\pi r l + \dfrac{1}{2}\pi r^2 + \dfrac{1}{2} \times 2r \times h$
a cone on a wider cylinder	the ring left showing: $\pi(R^2 - r^2)$
a pyramid on a cuboid	no top face of the cuboid (it is covered)
a prism with a sector cross-section	volume $= \dfrac{\theta}{360^\circ}\pi R^2 \times$ length

📖 Worked example

The diagram shows a pyramid with a square base of side $x$ cm. Its apex is $3$ cm vertically above the centre of the base, and its volume is $16$ cm³. Find $x$ .

Step 1 of 3

Volume = ⅓ × base area × height: write an equation in x

\dfrac{1}{3} \times x^2 \times 3 = 16

📖 Worked example

The diagram shows a solid that is half of a right circular cone, cut along a plane through its axis. Its base radius is $3$ cm and its volume is $6\pi$ cm³. Find its height, $h$ cm.

Step 1 of 2

Half a cone: volume = ½ × ⅓πr²h. Write an equation in h

\dfrac{1}{2} \times \dfrac{1}{3} \times \pi \times 3^2 \times h = 6\pi

📖 Worked example

A solid cone has base radius $r$ cm and height $h$ cm. Its volume is equal to the volume of a solid sphere of radius $r$ cm. Find $h$ in terms of $r$ . Hence show that the total surface area of the cone is $k\pi r^2$ cm², and find $k$ , leaving your answer in surd form.

Step 1 of 3

Equal volumes: ⅓πr²h = the other volume. Divide both sides by ⅓πr²

\dfrac{1}{3}\pi r^2 h = \dfrac{4}{3}\pi r^3

h = 4r

📖 Worked example

The diagram shows a solid made of a cylinder of radius $4$ cm and height $5$ cm, with a cone of base radius $3$ cm and height $4$ cm standing on the centre of its top face. Find the volume of the solid, correct to 3 significant figures.

Step 1 of 3

Volume of the cylinder = πR²H

\pi \times 4^2 \times 5 = 80\pi

📖 Worked example

A solid is made of a cylinder of radius $r$ cm and height $r$ cm standing on the flat face of a hemisphere of radius $2r$ cm, with their centres in line. Find the total surface area of the solid in the form $k\pi r^2$ cm², where $k$ is a number.

Step 1 of 4

Curved surface of the hemisphere: 2π(2r)²

2\pi(2r)^2 = 8\pi r^2

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Sec 2 · G3 (Express)

Pyramids, cones and spheres

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

Start free Worksheet (PDF)

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.

📖 Worked example

Step 1 of 3

Half the diagonal of the base: O is the centre, so OB is half of BD (Pythagoras on the base)

OB^2 = 3^2 + 3^2

= 18

📖 Worked example

Step 1 of 2

The arc of the sector becomes the circumference of the cone's base

\dfrac{120}{360} \times 2\pi \times 6 = 2\pi r

r = 2

📖 Worked example

A solid sphere of radius $r$ cm has the same volume as a solid cone of base radius $\dfrac{3}{2}r$ cm and height $h$ cm. Express $h$ in terms of $r$ .

Step 1 of 3

Volume of the sphere: ⁴⁄₃πr³

Volume

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

📖 Worked example

A solid metal cylinder of radius $6$ cm and height $10$ cm is melted down and recast into small solid spheres, each of radius $0.8$ cm. Find the greatest number of spheres that can be made.

Step 1 of 3

The volume of metal (melting doesn't change it)

Volume of the cylinder

= \pi \times 6^2 \times 10

= 1130.9733\ldots

cm³

📖 Worked example

The curved surface of a hemispherical dome of radius $6$ m is to be painted. One tin of paint covers $8$ m². Find the number of tins of paint that must be bought.

Step 1 of 3

Area of the curved surface of the dome: 2πr²

Area

= 2 \times \pi \times 6^2

= 226.1946\ldots

m²

Solid	Surface to count
half a cone (cut through its axis)	$\dfrac{1}{2}\pi r l + \dfrac{1}{2}\pi r^2 + \dfrac{1}{2} \times 2r \times h$
a cone on a wider cylinder	the ring left showing: $\pi(R^2 - r^2)$
a pyramid on a cuboid	no top face of the cuboid (it is covered)
a prism with a sector cross-section	volume $= \dfrac{\theta}{360^\circ}\pi R^2 \times$ length

📖 Worked example

The diagram shows a pyramid with a square base of side $x$ cm. Its apex is $3$ cm vertically above the centre of the base, and its volume is $16$ cm³. Find $x$ .

Step 1 of 3

Volume = ⅓ × base area × height: write an equation in x

\dfrac{1}{3} \times x^2 \times 3 = 16

📖 Worked example

The diagram shows a solid that is half of a right circular cone, cut along a plane through its axis. Its base radius is $3$ cm and its volume is $6\pi$ cm³. Find its height, $h$ cm.

Step 1 of 2

Half a cone: volume = ½ × ⅓πr²h. Write an equation in h

\dfrac{1}{2} \times \dfrac{1}{3} \times \pi \times 3^2 \times h = 6\pi

📖 Worked example

Step 1 of 3

Equal volumes: ⅓πr²h = the other volume. Divide both sides by ⅓πr²

\dfrac{1}{3}\pi r^2 h = \dfrac{4}{3}\pi r^3

h = 4r

📖 Worked example

Step 1 of 3

Volume of the cylinder = πR²H

\pi \times 4^2 \times 5 = 80\pi

📖 Worked example

Step 1 of 4

Curved surface of the hemisphere: 2π(2r)²

2\pi(2r)^2 = 8\pi r^2

Practise this →

Try a question: Pyramids, cones and spheres

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

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.

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