The circle: definitions and terms

A circle is the locus of all the points in the plane whose distance from one point, the centre, is a fixed number $\displaystyle R$.

• A radius is a segment from the centre to a point on the circle. All the radii of a circle are equal, and their length is $\displaystyle R$.
• A chord is a segment joining two points on the circle – $\displaystyle BC$ in the drawing.
• A diameter is a chord through the centre – $\displaystyle DE$ in the drawing. Its length is two radii: $\displaystyle 2R$.

So, for any point $\displaystyle P$:

• $\displaystyle P$ is on the circle if $\displaystyle {OP}={R}$;
• $\displaystyle P$ is inside the circle if $\displaystyle OP<R$;
• $\displaystyle P$ is outside the circle if $\displaystyle OP>R$.

Two points on a circle split it into two arcs. A central angle is an angle whose vertex is the centre of the circle and whose two arms are radii.

• The central angle $\displaystyle \angle AOB$ stands on the arc $\displaystyle AB$ inside it.
• The measure of an arc is the central angle standing on it. In the drawing the arc $\displaystyle AB$ inside the angle is $\displaystyle 70^{\circ}$.

The chord $\displaystyle AB$ splits the circle into a minor arc – the one the central angle $\displaystyle \angle AOB$ stands on – and a major arc, all the rest: $\displaystyle {360^{\circ}-70^{\circ}}={290^{\circ}}$.

An inscribed angle is an angle whose vertex is on the circle and whose two arms are chords. You will meet it in the lesson “The inscribed angle”.

A disc is the circle together with all the points inside it: the circle is the line, the disc is the area the line encloses.

• A sector is a part of a disc between two radii and the arc between them – like a slice of pizza.
• A sector has a central angle. A sector of $\displaystyle 90^{\circ}$ is a quarter of the disc, since $\displaystyle {\frac{90}{360}}={\frac{1}{4}}$.

On the axes, the distance of a point from the centre is found by the formula for the distance between two points, and compared with the radius.

Example: the centre of the circle is $\displaystyle M(1,2)$, and the radius is $\displaystyle 5$.
Is the point $\displaystyle P(4,6)$ on the circle?

$\displaystyle {MP}={R}$, so $\displaystyle P$ is on the circle.

$\displaystyle O$ is the centre of the circle, and $\displaystyle A$, $\displaystyle B$ are points on it.
$\displaystyle {\angle AOB}={124^{\circ}}$.

The points $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ on a circle split it into three arcs: $\displaystyle AB$, $\displaystyle BC$ and $\displaystyle CA$.
The arc $\displaystyle AB$ is $\displaystyle 100^{\circ}$, and the arc $\displaystyle BC$ is $\displaystyle 150^{\circ}$.
How many degrees is the arc $\displaystyle CA$?

$\displaystyle O$ is the centre of the circle, and $\displaystyle CD$ is a diameter.
$\displaystyle AB$ is a chord that does not pass through $\displaystyle O$.
Prove: $\displaystyle AB<CD$ – that is, the diameter is the longest chord.

The centre of the circle is $\displaystyle M(2,-1)$, and the radius is $\displaystyle 5$.
The points: $\displaystyle P(5,3)$, $\displaystyle Q(-1,2)$, $\displaystyle S(7,2)$.

Which point is on the circle, which inside it and which outside it? Explain.