Circles
Theory
A circle is the set of all points the same distance — the radius \(r\) — from a centre. The two key formulas both use \(\pi \approx 3.14159\): circumference \(C = 2\pi r\) (or \(\pi d\)) and area \(A = \pi r^2\). A sector (pizza slice) uses the fraction \(\theta/360\) of the full circle's circumference or area.
A circle is the set of all points the same distance from a fixed centre. That distance is the radius \(r\). Straight across through the centre is the diameter \(d = 2r\).
Two key formulas both use the constant \(\pi \approx 3.14159\). The circumference (distance around) is \(C = 2\pi r\) — or equivalently \(C = \pi d\). The area is \(A = \pi r^2\).
A sector is a "pizza slice" of a circle — the region between two radii and the arc joining them. If the angle at the centre is \(\theta\) degrees, the slice is the fraction \(\theta/360\) of the whole circle. A semicircle is \(\theta = 180\) (half); a quarter circle is \(\theta = 90\).
Circumference (distance around) of a circle:
Area of a circle:
Arc length of a sector with central angle \(\theta\) (in degrees):
Sector area:
Annulus (ring) area:
How to solve a circle problem
- Identify what you need — circumference, area, sector, or annulus.
- Check the input: if you're given the diameter, halve it to get \(r\) before using area or sector formulas.
- For a sector, multiply the full circle formula by the fraction \(\theta/360\).
- For an annulus, subtract the inner circle's area from the outer circle's area.
- For a rolling problem, find the circumference first; total distance is \(n \times C\) for \(n\) revolutions.
Apply both formulas with \(r = 7\).
| \(C\) | \(=\) | \(2\pi \times 7\) |
| \(C\) | \(\approx\) | \(43.98\) cm |
| \(A\) | \(=\) | \(\pi \times 7^2\) |
| \(A\) | \(\approx\) | \(153.94\) cm² |
Circumference \(\approx \textbf{43.98}\) cm; area \(\approx \textbf{153.94}\) cm².
Use \(\theta/360\) of each whole-circle formula.
| \(\ell\) | \(=\) | \(\dfrac{72}{360} \times 2\pi \times 10\) |
| \(\ell\) | \(=\) | \(0.2 \times 20\pi\) |
| \(\ell\) | \(\approx\) | \(12.57\) cm |
| \(A\) | \(=\) | \(\dfrac{72}{360} \times \pi \times 10^2\) |
| \(A\) | \(\approx\) | \(62.83\) cm² |
Arc length \(\approx \textbf{12.57}\) cm; sector area \(\approx \textbf{62.83}\) cm².
Find the circumference, multiply by revolutions, then convert.
| \(C\) | \(=\) | \(\pi \times 60\) |
| \(C\) | \(\approx\) | \(188.50\) cm |
| \(\text{distance}\) | \(=\) | \(150 \times 188.50\) |
| \(\text{distance}\) | \(\approx\) | \(28{,}274.33\) cm |
| \(\text{distance}\) | \(\approx\) | \(282.74\) m |
The bike travels about \(\textbf{282.74}\) m.
Outer area minus inner area.
| \(A\) | \(=\) | \(\pi \times 5^2 - \pi \times 3^2\) |
| \(A\) | \(=\) | \(25\pi - 9\pi = 16\pi\) |
| \(A\) | \(\approx\) | \(50.27\) m² |
The garden area is about \(\textbf{50.27}\) m².
Common pitfalls
Frequently asked questions
What is the formula for the circumference of a circle?
\(C = 2\pi r\), or equivalently \(C = \pi d\) where \(d\) is the diameter.
What is the formula for the area of a circle?
\(A = \pi r^2\). If given the diameter, halve it to get \(r\) first.
What is the difference between radius and diameter?
Radius is the distance from the centre to the edge. Diameter is straight across through the centre. \(d = 2r\).
How do you find the arc length of a sector?
\(\ell = \dfrac{\theta}{360} \times 2\pi r\) — the fraction \(\theta/360\) of the full circumference.
How do you find the area of a sector?
\(A = \dfrac{\theta}{360} \times \pi r^2\) — the fraction \(\theta/360\) of the full circle's area.
How do you find the area of an annulus or ring?
\(A = \pi R^2 - \pi r^2\), where \(R\) is the outer radius and \(r\) is the inner radius.
Video Lesson
- Area and Perimeter: Circles Watch
Practice Questions
12 questions available.
Practice Questions