Circles, Sectors and Circle Theorems

Every circle has these key parts:

Term	Definition
Centre	The fixed point equidistant from all points on the circle
Radius	A line from the centre to the circumference ($r$)
Diameter	A chord through the centre; $d=2r$
Chord	A straight line joining two points on the circumference
Arc	A portion of the circumference
Sector	A "pie slice" — region bounded by two radii and an arc
Segment	Region between a chord and the arc it cuts off
Tangent	A straight line that touches the circle at exactly one point
Circumference	The complete perimeter of the circle

Key property: a tangent at a point is perpendicular to the radius drawn to that point.

A sector with angle $\theta °$ (out of $360°$) takes a fraction $\frac{\theta }{360}$ of the full circle.

$\text{Arc length}=\frac{\theta }{360}\times 2\pi r$

$\text{Sector area}=\frac{\theta }{360}\times \pi r^2$

Worked example — sector with radius 8 cm and angle $135°$:

Arc length $=\frac{135}{360}\times 2\pi \times 8=\frac{3}{8}\times 16\pi =6\pi \approx 18.85$ cm

Sector area $=\frac{135}{360}\times \pi \times 64=\frac{3}{8}\times 64\pi =24\pi \approx 75.40$ cm² ✓

Perimeter of a sector = arc length + 2 radii $=6\pi +16\approx 34.85$ cm ✓

Working backwards — find the angle given arc length $=10\pi$ cm and $r=15$ cm:

$10\pi =\frac{\theta }{360}\times 30\pi ⟹\theta =\frac{10\pi \times 360}{30\pi }=120°$ ✓

Theorem 1 — Angle at the centre:

The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.

$\angle AOB=2\times \angle ACB$ (where $O$ is the centre, $A$, $B$, $C$ on the circle)

Theorem 2 — Angle in a semicircle:

An angle in a semicircle (inscribed in a diameter) is a right angle. $\angle ACB=90°$ when $AB$ is a diameter.

Theorem 3 — Angles in the same segment:

Angles subtended by the same chord on the same side of the chord are equal.

$\angle ACB=\angle ADB$ (both in the same segment)

Theorem 4 — Opposite angles in a cyclic quadrilateral:

Opposite angles of a cyclic quadrilateral (all four vertices on the circle) sum to $180°$.

$\angle A+\angle C=180°$; $\angle B+\angle D=180°$

Theorem 5 — Tangent-radius perpendicularity:

A radius to the point of tangency is perpendicular to the tangent. If $OT$ is the radius drawn to the tangent point $T$, then $OT\perp$ tangent.

Theorem 6 — Two tangents from an external point:

The two tangents drawn from an external point to a circle are equal in length.

$PA=PB$ where $P$ is external, $A$ and $B$ are the tangent points.

Theorem 7 — Perpendicular from centre to chord:

The perpendicular from the centre of a circle to a chord bisects the chord.

Theorem 8 — Alternate segment theorem:

The angle between a tangent and a chord at the point of tangency equals the inscribed angle subtending the same chord on the opposite side.

$\angle$ (tangent-chord) $=\angle$ (inscribed angle in alternate segment)

Worked example — $O$ is the centre of a circle. Chord $AB$ makes an angle of $35°$ with the tangent at $A$. Find the angle $\angle ACB$ in the alternate segment.

By the alternate segment theorem: $\angle ACB=35°$ ✓

For proof questions (Higher), the method is:

1. Mark relevant angles using a letter.
2. Apply named theorems step by step.
3. State the conclusion and the theorem used at each step.

Worked example — prove that the angle subtended by a diameter at the circumference is $90°$.

Let $AB$ be a diameter, $O$ the centre, $C$ a point on the circle. Let $\angle ACB=x$.

By the angle at the centre theorem: $\angle AOB=2x$. But $\angle AOB=180°$ (straight line, since $AB$ is a diameter).

Therefore $2x=180°⟹x=90°$ ✓

Worked example — in a cyclic quadrilateral $ABCD$, $\angle DAB=110°$. Find $\angle BCD$.

Opposite angles sum to $180°$: $\angle BCD=180°-110°=70°$ ✓

1. Sector perimeter — omitting the two radii

Sector perimeter $=$ arc length $+2r$. Many students calculate only the arc length, forgetting the two straight sides (the radii).

2. Circle theorem — applying the angle at the centre rule incorrectly

The central angle is double the inscribed angle only when both are subtended by the same arc. Confusing which arc each angle is subtended by leads to errors.

3. Alternate segment theorem — picking the wrong angle

The alternate segment theorem connects a tangent-chord angle with the inscribed angle in the segment on the opposite side of the chord. Take care to identify which segment is the "alternate" one.

4. Cyclic quadrilateral — assuming all adjacent angles sum to 180°

It is the opposite angles (not adjacent) that are supplementary in a cyclic quadrilateral.

Mistake	Correction
"Sector perimeter with $r=5$, $\theta =90°$: $\frac{1}{4}\times 2\pi \times 5=\frac{5\pi }{2}$"	Add the two radii: perimeter $=\frac{5\pi }{2}+10\approx 17.85$ cm
"Angle at circumference $=40°$, so angle at centre $=20°$"	Angle at centre is twice the circumference angle: $80°$
"In cyclic quad, adjacent angles sum to $180°$"	Opposite angles sum to $180°$