Circumference and area of a circle

How far is it round a circle?

Take a tin lid and a piece of string. Wrap the string once round the lid. Straighten the string and measure it. Then measure across the lid, through the centre. The string is a little more than three times as long as the width of the lid. This happens for every circle, big or small.

• The distance round a circle is its circumference. Some books call it the perimeter of the circle. It is the same length.
• The distance across, through the centre, is the diameter $d$. It is twice the radius $r$.
• The circumference divided by the diameter is the same number for every circle. We call that number $\pi$ (pi).

What is the value of $\pi$? It is a number that never ends: $3.14159\ldots$. Two short forms are common. One is $3.14$. The other is $\dfrac{22}{7} = 3.142857\ldots$. Both are a little different from $\pi$. The fraction $\dfrac{22}{7}$ is larger than $\pi$ by about $0.04\%$, which is 4 parts in 10 000.

• If a question says "take $\pi = \dfrac{22}{7}$", use it and give a number. This is easy when the radius is a multiple of $7$, because the $7$ cancels.
• If a question gives no value, leave the answer in terms of $\pi$, such as $50\pi\ \text{cm}^2$, or use the $\pi$ button of a calculator.
• In this chapter we show the exact form with $\pi$ as well as the number, so you can see both.

Where books differ. Indian board papers usually say "take $\pi = \dfrac{22}{7}$", and sometimes "$3.14$". GCSE and IGCSE papers say "leave your answer in terms of $\pi$" or "give your answer to 3 significant figures". SAT Math prints the formulas at the start of the section. The method is the same in every case.

The area of a circle

Cut a circle into thin slices, like a pizza. Lay the slices side by side, one pointing up and the next pointing down. They fit together into a shape that is nearly a rectangle. Its height is the radius $r$. Its length is half the circumference, which is $\pi r$. The thinner the slices, the closer the shape is to a true rectangle.

So the area is the length times the height:

Example 1. A circle has a radius of $14\ \text{cm}$. Find its circumference and its area. Take $\pi = \dfrac{22}{7}$.

1. The circumference. $C = 2\pi r = 2 \times \dfrac{22}{7} \times 14 = 88\ \text{cm}$.
2. The area. $A = \pi r^2 = \dfrac{22}{7} \times 14 \times 14 = 616\ \text{cm}^2$.

Example 2. The circumference of a circle is $66\ \text{cm}$. Find its radius and its area. Take $\pi = \dfrac{22}{7}$.

1. The radius. $2\pi r = 66$, so $r = \dfrac{66}{2\pi} = \dfrac{66 \times 7}{2 \times 22} = 10.5\ \text{cm}$. We want $r$, so we divide.
2. The area. $A = \dfrac{22}{7} \times 10.5 \times 10.5 = 346.5\ \text{cm}^2$.

Example 3. The area of a circle is $154\ \text{cm}^2$. Find its circumference. Take $\pi = \dfrac{22}{7}$.

1. The radius. $\pi r^2 = 154$, so $r^2 = 154 \times \dfrac{7}{22} = 49$. Then $r = 7\ \text{cm}$.
2. The circumference. $C = 2 \times \dfrac{22}{7} \times 7 = 44\ \text{cm}$.

Example 4, in terms of $\pi$. A circle has a radius of $5\ \text{cm}$. Then $C = 2\pi \times 5 = 10\pi\ \text{cm}$ and $A = \pi \times 5^2 = 25\pi\ \text{cm}^2$. These are exact. With a calculator, $10\pi = 31.4\ \text{cm}$ and $25\pi = 78.5\ \text{cm}^2$.

What happens when the radius changes?

The circumference has $r$ in it once. The area has $r$ twice. So if the radius becomes $2$ times as large, the circumference becomes $2$ times as large and the area becomes $2^2 = 4$ times as large. If the radius becomes $3$ times as large, the area becomes $9$ times as large.

This is a general rule: when every length grows $k$ times, every area grows $k^2$ times.

Lengths in different units

Many problems mix centimetres and metres. Follow three rules.

1. Use one unit. Change every length to the same unit before you calculate.
2. An area unit is a square. $1\ \text{m} = 100\ \text{cm}$, so $1\ \text{m}^2 = 100 \times 100 = 10\,000\ \text{cm}^2$.
3. Answer in the unit asked for. Write the unit with the answer.

Example 5. A round table top has a radius of $70\ \text{cm}$. Find its area in square metres. Take $\pi = \dfrac{22}{7}$.

1. Method 1: change the radius first. $70\ \text{cm} = 0.7\ \text{m}$. So $A = \dfrac{22}{7} \times 0.7 \times 0.7 = 1.54\ \text{m}^2$.
2. Method 2: change the answer. $A = \dfrac{22}{7} \times 70 \times 70 = 15\,400\ \text{cm}^2$. Then $15\,400 \div 10\,000 = 1.54\ \text{m}^2$.

Both methods agree. A common slip is to divide by $100$ instead of $10\,000$ in Method 2.

Example 6, a cost. A round pond has a diameter of $28\ \text{m}$. A fence round it costs $15$ rupees for each metre. The radius is $14\ \text{m}$, so the fence is $2 \times \dfrac{22}{7} \times 14 = 88\ \text{m}$ long. The cost is $88 \times 15 = 1320$ rupees.

Practise with new numbers

Try it in Python

A program can compute the circumference and the area for many circles at once. This one uses the exact $\pi$ and also $\dfrac{22}{7}$, and prints how far apart the two areas are.

Common mistakes

• Using the diameter as the radius. Check which one the question gives. The radius is half of the diameter.
• Forgetting to square. The area is $\pi r^2$, not $\pi r$. The second one is half the circumference.
• Mixing up the circumference and the area. The circumference is a length, in cm or m. The area is in $\text{cm}^2$ or $\text{m}^2$.
• Dividing by 100 for an area. A square metre is $10\,000$ square centimetres.

Remember these

• $C = 2\pi r = \pi d$ and $A = \pi r^2$.
• $\pi$ is $C \div d$ for every circle. Use $\dfrac{22}{7}$ only when told, and show the exact form with $\pi$.
• Doubling the radius doubles the circumference and makes the area four times as large.
• Change all lengths to one unit first, and remember $1\ \text{m}^2 = 10\,000\ \text{cm}^2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A circle has a diameter of $14\ \text{cm}$. Take $\pi = \dfrac{22}{7}$. Find its circumference and its area.

2. The circumference of a circle is $110\ \text{cm}$. Take $\pi = \dfrac{22}{7}$. Find its radius.

3. A circle has a radius of $6\ \text{cm}$. Write its circumference and its area in terms of $\pi$.

4. A circle has a radius of $35\ \text{cm}$. Take $\pi = \dfrac{22}{7}$. Find its area in square metres.

5. A circle has a radius $4$ times as large as another circle. How many times as large is its area?