Circles: area, arcs, sectors and equations

Find circumference, area, arc length and sector area, and read the equation of a circle.

Also known as: circle area, arc length, sector area, equation of a circle.

SAT · Geometry and Trigonometry GCSE Maths · Geometry and measures IGCSE Maths · Mensuration

Key ideas

Circumference $= 2\pi r = \pi d$ ; area $= \pi r^2$ .
Arc length $= \frac{\theta}{360} \times 2\pi r$ ; sector area $= \frac{\theta}{360} \times \pi r^2$ .
Circle with centre $(h, k)$ and radius $r$ : $(x - h)^2 + (y - k)^2 = r^2$ .
Complete the square to find the centre and radius from an expanded equation.

Circumference and area

For radius $r$ : circumference $C = 2\pi r$ and area $A = \pi r^2$ . A circle with radius $5$ has area $25\pi$ . Leave answers "in terms of $\pi$ " when asked; otherwise round at the end.

Arcs and sectors

An arc or sector is a fraction $\frac{\theta}{360}$ of the full circle. With $r = 9$ and $\theta = 80°$ : $\text{arc} = \frac{80}{360} \times 18\pi = 4\pi.$ With $r = 6$ and $\theta = 120°$ : sector area $= \frac{1}{3} \times 36\pi = 12\pi$ .

Equation of a circle

$(x - h)^2 + (y - k)^2 = r^2$ has centre $(h, k)$ and radius $r$ . Note the signs: $(x - 2)^2 + (y + 3)^2 = 16$ has centre $(2, -3)$ and radius $4$ .

From $x^2 + y^2 - 6x + 8y = 0$ , complete the square: $(x - 3)^2 + (y + 4)^2 = 25$ , so the radius is $5$ .

Worked examples

Example 1 · Sector area

Find the area of a sector with radius $6$ and angle $120°$ , in terms of $\pi$ .

Step 1.Fraction of circle: $\frac{120}{360} = \frac{1}{3}$ .
Step 2. $\frac{1}{3} \times \pi \times 6^2 = 12\pi$ .

Answer: $12\pi$

Example 2 · Centre and radius

Find the radius of $x^2 + y^2 - 6x + 8y = 0$ .

Step 1. $(x^2 - 6x + 9) + (y^2 + 8y + 16) = 25$ .
Step 2. $(x - 3)^2 + (y + 4)^2 = 25$ , so $r = 5$ .

Answer: 5

Common mistakes

Using the diameter in $\pi r^2$ .
Reading the centre of $(x - 2)^2 + (y + 3)^2 = 16$ as $(-2, 3)$ .
Giving $r^2$ as the radius (e.g. $16$ instead of $4$ ).

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Circles: area, arcs, sectors and equationsEasier

What is the area of a circle with radius 5?

Question 2Circles: area, arcs, sectors and equationsEasier

What is the circumference of a circle with diameter 10? Give your answer to 1 decimal place.