# Circle Geometry: Chords and Angles

> Circle words, chord properties, the angle at the centre, angles in the same segment and in a semicircle, and cyclic quadrilaterals.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/circle-geometry/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The broken plate
An archaeologist digs up a piece of an old round plate. She wants to know how big the whole plate was. She draws two chords across the broken piece, then draws the perpendicular bisector of each chord. The two lines cross at one point: the centre of the plate.

That trick works because of a circle theorem: **the perpendicular bisector of any chord passes through the centre.** Circles hide many facts like this, and each one turns a tricky angle question into a one-line answer.

**In short:** Circle geometry is a set of rules about chords and angles in a circle. The angle at the centre is twice the angle at the circumference on the same arc, the angle in a semicircle is $90^\circ$, and opposite angles of a cyclic quadrilateral add to $180^\circ$.

## Starting point: Years 9 and 10

### Circle words

- **Radius:** centre to the edge. **Diameter:** right across through the centre.
- **Chord:** a straight line joining two points on the circle.
- **Arc:** part of the circumference. A chord splits the circle into a **minor** and a **major** segment.
- **Subtend:** a chord or arc "makes" an angle at a point. Angle $APB$ is subtended by chord $AB$.
- A **cyclic quadrilateral** has all four corners on the circle.

### Chord properties

- A line from the centre **perpendicular** to a chord **bisects** it.
- Chords of equal length are the same distance from the centre.
- Equal chords subtend equal angles at the centre.

### Worked example: Using Pythagoras with a chord
A chord of length 16 cm is 6 cm from the centre. The perpendicular cuts it into 8 and 8, so the radius is $\sqrt{8^2 + 6^2} = 10$ cm.

### Angle theorems

*Diagram: Circle with centre O. The angle at the centre AOB is twice the angle APB at the circumference*
*The angle at the centre is twice the angle at the circumference on the same arc.*

### Key idea: Five angle theorems
1. The angle at the **centre** is **twice** the angle at the circumference standing on the same arc.
2. Angles at the circumference standing on the same arc are **equal** (angles in the same segment).
3. The angle in a **semicircle** is $90^\circ$.
4. **Opposite angles** of a cyclic quadrilateral add to $180^\circ$.
5. The **exterior angle** of a cyclic quadrilateral equals the interior opposite angle.

Theorem 3 is theorem 1 with a straight angle: $180^\circ$ at the centre gives $90^\circ$ at the edge.

### Worked example: Cyclic quadrilateral
In cyclic quadrilateral $ABCD$, $\angle A = 75^\circ$ and $\angle B = 110^\circ$. Then $\angle C = 180^\circ - 75^\circ = 105^\circ$ and $\angle D = 180^\circ - 110^\circ = 70^\circ$.

### Exam tip: Look for isosceles triangles
Two radii make an isosceles triangle with the centre, so its base angles are equal. Mark every radius you can see.

### Watch out: Is the angle at the centre?
Theorem 1 needs one angle **at the centre**. A point that only looks central is not enough. Check for the letter $O$.

**Mind map**

```text
Circle geometry
  Words
    Radius, diameter, chord
    Arc, segment, subtend
  Chords
    Perpendicular from centre bisects
    Equal chords, equal distance
  Angles
    Centre is twice circumference
    Same segment are equal
    Semicircle is 90°
  Cyclic quadrilateral
    Opposite angles add to 180°
    Exterior equals interior opposite
```

### Year 11: Circles on the number plane
On the number plane, a circle with centre $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$.

The chord theorem becomes coordinate geometry: the perpendicular bisector of a chord passes through the centre. For the chord from $A(-2, 0)$ to $B(4, 0)$, the midpoint is $(1, 0)$ and the perpendicular bisector is the line $x = 1$, so the centre lies on $x = 1$.

The angle in a semicircle gives a gradient test: if $AB$ is a diameter and $P$ is on the circle, then $m_{PA} \times m_{PB} = -1$.

### Year 12: The biggest rectangle in a circle
A rectangle sits inside a circle of radius 5, with its corners on the circle. Its diagonal is a diameter (angle in a semicircle), so a rectangle of width $x$ has height $\sqrt{100 - x^2}$. Its area is

$$A = x\sqrt{100 - x^2}$$

Calculus gives the maximum at $x = \sqrt{50}$: a square with area 50. The geometry fact sets up the function, and calculus finds the best value.

## Quick summary

- A perpendicular from the centre bisects a chord. Use Pythagoras with half the chord.
- The angle at the centre is twice the angle at the circumference on the same arc.
- Angles in the same segment are equal. The angle in a semicircle is $90^\circ$.
- Opposite angles of a cyclic quadrilateral add to $180^\circ$.
- Two radii form an isosceles triangle.

## Quiz

Try these questions from the chapter quiz.

1. A line joining two points on a circle is called:
   - A. a radius
   - B. a chord
   - C. a tangent
   - D. an arc
   - Answer: B. a chord. A chord's ends are both on the circle.

2. A chord is 24 cm long and 5 cm from the centre. The radius is:
   - A. 12 cm
   - B. 13 cm
   - C. 17 cm
   - D. 29 cm
   - Answer: B. 13 cm. $\sqrt{12^2 + 5^2} = 13$.

3. The angle at the circumference is $35^\circ$. The angle at the centre on the same arc is:
   - A. $35^\circ$
   - B. $17.5^\circ$
   - C. $70^\circ$
   - D. $145^\circ$
   - Answer: C. $70^\circ$. The centre angle is twice the circumference angle.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/circle-geometry/#quiz