Area of Circles
Every question with a fully worked solution.
- Area of Circles - Video - Area of Circles 1 Watch
Theory
The area of a circle is \(A = \pi r^2\): square the radius first, then multiply by \(\pi\). A circle with diameter \(12\) cm has \(r = 6\) cm, so \(A = 36\pi \approx 113.10~\text{cm}^2\). This Year 8 maths page shows how.
\(A = \pi r^2\), where \(r\) is the radius. Work out \(r^2 = r \times r\) first, then multiply by \(\pi\) (\(\pi \approx 3.14159\)).
Where it comes from. A circle fits inside a square of side \(2r\), area \(4r^2\), and fills a bit more than three quarters of it: about \(3.14r^2\). Or cut the circle into many equal sectors and lay them head to tail: they make a shape close to a parallelogram with base half the circumference, \(\pi r\), and height \(r\), so \(A = \pi r \times r = \pi r^2\).
Given the diameter? Halve it first: \(r = \dfrac{d}{2}\).
Exact or rounded. An exact area keeps \(\pi\); otherwise use the \(\pi\) key and round as asked. Area is in square units.
A ring (annulus) is the big circle take away the small circle: \(A = \pi R^2 - \pi r^2\). When a circle is cut out of another shape, subtract; when it is added on, add.
Working backwards: \(r^2 = A \div \pi\), so \(r = \sqrt{A \div \pi}\). Scaling: a radius \(3\) times as long gives an area \(3^2 = 9\) times as large.
Area of a circle:
How to find the area of a circle
- Find the radius (halve a diameter).
- Square it, then multiply by \(\pi\).
- Give an exact answer, or round at the end, in square units.
| \(A\) | \(=\) | \(\pi \times 11^2 = 121\pi\) |
\(121\pi~\text{cm}^2\)
| \(r\) | \(=\) | \(12 \div 2 = 6\) |
| \(A\) | \(=\) | \(\pi \times 6^2 = 36\pi\) |
| \(\) | \(=\) | \(113.097\ldots\) |
\(113.10~\text{cm}^2\)
(i) in terms of \(\pi\)
(ii) to \(2\) decimal places
| \(A\) | \(=\) | \(\pi \times 8^2 = 64\pi\) |
| \(\) | \(=\) | \(201.06\ldots\) |
(i) \(64\pi~\text{cm}^2\)
(ii) \(201.06~\text{cm}^2\)
(i) the area of the trapdoor
(ii) the area of deck left
each to \(2\) decimal places.
| \(A_1\) | \(=\) | \(\pi \times 1.2^2 = 4.523\ldots\) |
| \(A\) | \(=\) | \(7.5 - 4.523\ldots = 2.976\ldots\) |
(i) \(4.52~\text{m}^2\)
(ii) \(2.98~\text{m}^2\)
| \(R\) | \(=\) | \(1.5 + 0.5 = 2\) |
| \(A\) | \(=\) | \(\pi \times 2^2 - \pi \times 1.5^2\) |
| \(\) | \(=\) | \(1.75\pi = 5.497\ldots\) |
\(5.50~\text{m}^2\)
(i) its radius
(ii) the length of padding around its edge
each to \(2\) decimal places.
| \(r^2\) | \(=\) | \(7 \div \pi = 2.228\ldots\) |
| \(r\) | \(=\) | \(1.4927\ldots\) |
| \(C\) | \(=\) | \(2\pi \times 1.4927\ldots = 9.378\ldots\) |
(i) \(1.49\) m
(ii) \(9.38\) m, using the unrounded radius.
Common pitfalls
Frequently asked questions
What is the formula for the area of a circle?
\(A = \pi r^2\).
How do I find the area from the diameter?
Halve the diameter to get the radius, then use \(A = \pi r^2\).
How do I find the radius from the area?
Divide the area by \(\pi\), then take the square root.
How do I find the area of a ring?
Big circle minus small circle: \(\pi R^2 - \pi r^2\).
If the radius is halved, what happens to the area?
It becomes \(\dfrac{1}{4}\) as large.