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Master: Volume and surface area

Measurement · step 4 of 5

A question to start

Mars is a little more than half as wide as Earth. So if Earth were a hollow ball, how many Mars-sized balls of rock would it take to fill it? Two? Four?

The answer is about six and a half. Halving the width of a ball does not halve the space inside it; it cuts it to about an eighth. Yet the surface of Mars, all its land put together, is almost exactly as big as all the dry land on Earth. Volume and surface area grow in very different ways, and this page shows how to calculate both.

The idea

Volume

Volume is the amount of space a solid object takes up. Just as area counts unit squares, volume counts unit cubes:

  • a cubic centimetre (cm³) is a cube 1 cm on each edge, about the size of a sugar cube;
  • a cubic metre (m³) is a cube 1 m on each edge, big enough for a person to crouch inside.

The little 3 is a reminder that volume has three directions: length, width and height.

Boxes and prisms

A box (a cuboid) 4 cm long, 3 cm wide and 2 cm high holds 2 layers, each of \(4 \times 3 = 12\) cubes: 24 cm³.

\[\text{volume of a cuboid} = \text{length} \times \text{width} \times \text{height}\]

A prism is any solid with the same shape all the way through, like a loaf of sliced bread: a triangular prism, a hexagonal prism. Each slice has the same area, so:

\[\text{volume of a prism} = \text{area of the end} \times \text{length}\]

A cylinder (the shape of a can) is a prism with a circle at each end. With radius \(r\) and height \(h\):

\[V = \pi r^2 h\]

Pyramids and cones

A pyramid rises from a flat base to a point. A cone is a pyramid with a circular base. Fill a cone with sand and pour it into a cylinder with the same base and height: it takes exactly three cones to fill the cylinder. The same is true for a pyramid and the prism it fits in. So:

\[V = \tfrac13 \times \text{area of the base} \times \text{height} \qquad \text{for a cone: } V = \tfrac13 \pi r^2 h\]

Spheres

A sphere is a perfect ball: every point on its surface is the same distance \(r\) from its centre.

\[V = \tfrac43 \pi r^3\]

Archimedes, about 2200 years ago, discovered that a sphere fills exactly two thirds of the smallest cylinder it fits inside. The story goes that he was so proud of this that he asked for a sphere inside a cylinder to be carved on his tomb.

Capacity: litres and cubic centimetres

Volume and capacity are linked exactly:

\[1 \text{ cm}^3 = 1 \text{ mL} \qquad 1000 \text{ cm}^3 = 1 \text{ L} \qquad 1 \text{ m}^3 = 1000 \text{ L}\]

A litre is the volume of a cube 10 cm on each edge. A cubic metre of water is 1000 litres and weighs about a tonne.

Surface area

The surface area of a solid is the total area of all its outside surfaces: how much paint it would take to cover it, or how much cardboard to make it. It is an area, so it is measured in cm² or m².

The easiest way to find it is to imagine unfolding the solid flat. That flat pattern is called a net.

  • A cuboid has 6 rectangular faces in 3 matching pairs: \(\;SA = 2(lw + lh + wh)\).
  • A cylinder unfolds into two circles and one rectangle. The rectangle's width is the height \(h\), and its length is the distance around the circle, \(2\pi r\). So \(\;SA = 2\pi r^2 + 2\pi r h\).
  • A sphere cannot be unfolded flat (which is why every flat map of a globe stretches something). Archimedes found its area too: exactly four times the area of a circle with the same radius,
\[SA = 4\pi r^2\]
length 2πr (once round the circle) height h r

Worked examples

1. A water tank. A cylindrical tank has radius 0.5 m and height 1.2 m. How many litres does it hold?

  1. \(V = \pi r^2 h = \pi \times 0.5^2 \times 1.2 = \pi \times 0.25 \times 1.2 = 0.3\pi \approx 0.942\) m³.
  2. Each m³ is 1000 L, so the tank holds about 942 litres.

2. Paint for a box. A box is 4 m by 3 m by 2 m. What area must be painted, outside, on all six faces?

  1. The three different faces: \(4 \times 3 = 12\), \(4 \times 2 = 8\), \(3 \times 2 = 6\) m².
  2. Each comes twice: \(SA = 2(12 + 8 + 6) = 52\) m².

3. Mars and Earth. Mars has an average radius of about 3390 km; Earth's is about 6371 km.

  1. Volumes: the \(\tfrac43\pi\) is the same for both, so the ratio of volumes is the ratio of the radii cubed: \(\left(\dfrac{3390}{6371}\right)^3 \approx 0.532^3 \approx 0.151\). Mars has about 15% of Earth's volume, so about \(1 \div 0.151 \approx 6.6\) Mars would fill Earth.
  2. Surface of Mars: \(4\pi r^2 = 4\pi \times 3390^2 \approx 144\) million km².
  3. Earth's dry land covers about 149 million km² (the oceans cover the rest of its 510 million km²). So the whole surface of Mars is a little less than all the land on Earth put together.

4. An ice-cream cone. A cone has radius 3 cm and height 10 cm. \(V = \tfrac13 \pi \times 3^2 \times 10 = 30\pi \approx 94.2\) cm³.

Try it

  1. Find the volume and surface area of a box 4 cm by 3 cm by 2 cm.
  2. A cylinder has radius 2 cm and height 5 cm. Find its volume and surface area, to one decimal place.
  3. A cone has radius 3 cm and height 4 cm. Find its volume.
  4. A ball has radius 6 cm. Find its volume and surface area, to one decimal place.
  5. A fish tank is 2 m long, 1.5 m wide and 0.8 m deep. How many litres does it hold?
  6. How many cubic centimetres are there in one cubic metre?
Answers
  1. \(V = 4 \times 3 \times 2 = 24\) cm³. \(SA = 2(12 + 8 + 6) = 52\) cm².
  2. \(V = \pi \times 2^2 \times 5 = 20\pi \approx 62.8\) cm³. \(SA = 2\pi \times 2^2 + 2\pi \times 2 \times 5 = 8\pi + 20\pi = 28\pi \approx 88.0\) cm².
  3. \(V = \tfrac13 \pi \times 3^2 \times 4 = 12\pi \approx 37.7\) cm³.
  4. \(V = \tfrac43 \pi \times 6^3 = \tfrac43 \pi \times 216 = 288\pi \approx 904.8\) cm³. \(SA = 4\pi \times 36 = 144\pi \approx 452.4\) cm².
  5. \(2 \times 1.5 \times 0.8 = 2.4\) m³, and each m³ is 1000 L, so 2400 L.
  6. A cubic metre is 100 cm by 100 cm by 100 cm: \(100 \times 100 \times 100 = 1\,000\,000\) cm³.

Common mistake

Forgetting the \(\tfrac13\), or using the slant. A cone's volume is a third of the cylinder around it, not the whole cylinder. And, as with triangles, the height of a cone or pyramid is measured straight up from the base to the tip, not along the sloping side. A quick check: a cone should always come out smaller than the cylinder with the same base and height.

Where it leads