GeometryGeometryHigh school

Circles

Circumference, area, arcs and sectors.

A circle is the set of points at a fixed distance, the radius $r$ , from a centre. The circumference is $C = 2\pi r$ and the area is $A = \pi r^2$ . The constant $\pi \approx 3.14159$ is the ratio of any circle's circumference to its diameter.

Arcs and sectors are fractions of the whole circle. A central angle of $\theta$ degrees cuts off $\frac{\theta}{360}$ of the circumference and of the area. In radians this becomes the tidy formulas $s = r\theta$ for arc length and $A = \frac{1}{2}r^2\theta$ for sector area.

Key terms

Area of a circle: Pi times the radius squared.
$A = \pi r^2$
Arc length: For a central angle in radians, arc length is radius times angle.
$s = r\theta$

Key ideas

Equation of a circle with centre $(h, k)$ and radius $r$ : $(x - h)^2 + (y - k)^2 = r^2$ .
A tangent line is perpendicular to the radius at the point of contact.
An inscribed angle is half the central angle that subtends the same arc.
Complete the square to find the centre and radius from an expanded equation.

Worked example

Find the centre and radius of $x^2 + y^2 - 6x + 4y - 12 = 0$ .

Group: $(x^2 - 6x) + (y^2 + 4y) = 12$ .
Complete squares: $(x - 3)^2 - 9 + (y + 2)^2 - 4 = 12$ .
So $(x - 3)^2 + (y + 2)^2 = 25$ .
Centre $(3, -2)$ , radius $5$ .

Answer: Centre $(3, -2)$ , radius $5$

Common mistakes

Using the diameter in $\pi r^2$ .
Reading the centre of $(x - 3)^2 + (y + 2)^2 = 25$ as $(-3, 2)$ .
Forgetting to take the square root of $r^2$ .

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Find the area of a circle with radius 6, to two decimals.
Show answer
$113.10$
2.Find the arc length of a $90^\circ$ arc on a circle of radius 8, to two decimals.
Show answer
$12.57$
3.A circle has circumference $10\pi$ . What is its radius?
Show answer
$5$

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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