Area of Circles
The area of a circle depends only on its radius: \(A = \pi r^2\). This NSW Year 8 maths subtopic (NESA outcome MA4‑ARE‑C‑01) shows where the formula comes from and uses it with a given radius or diameter, giving answers in terms of \(\pi\) or rounded with the \(\pi\) key.
You will find the area of a ring such as a washer, the region left when a circle is cut from a rectangle, and a radius or diameter from a given area or circumference. You will also see why doubling the radius makes the area four times as large, decide whether one large pizza beats two smaller ones, and find how much of a lawn a sprinkler misses. It is part of Year 8 Area and Perimeter.
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.