What Is a Circle?

Circle

A circle is the collection of all points that are the same distance, called the radius, from a fixed centre.

Arc vs Chord

https://cdn.mathpix.com/cropped/2025_07_01_a141ee9b5a6f03e0cd28g-02.jpg?height=804&width=804&top_left_y=357&top_left_x=877

Identify the highlighted arc (curved) and chord (straight).

Parts of a Circle

Today we focus on two parts of a circle.

Learn to tell an arc from a chord in any diagram.

Key Points:

  • Arc – curved portion of the circumference.
  • Chord – straight line linking two points on the circle.

Multiple Choice Question

Question

The diameter of a circle is 10 cm. Find its radius.

A
5 cm
B
10 cm
C
15 cm
D
20 cm

Hint:

Divide the diameter by 2 to get the radius.

Formula for Arc Length

Arc length diagram demonstrating central angle θ

Arc highlighted by central angle θ

Arc Length Formula

Arc length is the distance along a circle's curved edge.

Use the formula below to write or identify the length of any arc.

Key Points:

  • \(L = 2\pi r \times \dfrac{\theta}{360^\circ}\)
  • \(r\) — radius of the circle
  • \( \theta \) — central angle in degrees
  • Substitute values to compute or verify arc length.

Worked Example

Goal: Calculate the length of a \(60^\circ\) arc on a 7 cm circle.

1

Note radius and angle

Radius \(r = 7\,\text{cm}\); central angle \(\theta = 60^\circ\).

2

Write arc formula

\(L = 2\pi r \times \frac{\theta}{360^\circ}\).

3

Substitute values

\(L = 2\pi(7)\times\frac{60}{360}\).

4

Simplify & calculate

\(\frac{60}{360} = \frac16\); \(L = 14\pi \times \frac16 = \frac{14\pi}{6} \approx 7.33\,\text{cm}\).

Pro Tip:

Keep \(\theta\) in degrees when using this version of the arc length formula.

Angle at Centre Rule

Angle at centre diagram

Central angle \(2x\) is twice circumference angle \(x\).

Angle at Centre Theorem

In a circle, the angle at the centre is twice the angle at the circumference made by the same chord.

Key Points:

  • \( \angle\text{centre} = 2 \times \angle\text{circumference} \)
  • Both angles subtend the same chord.
  • Applies to minor and major arcs alike.

Multiple Choice Question

Question

The angle at the centre of a circle is 100°. What is the corresponding angle at the circumference?

1
25°
2
50°
3
100°
4
200°

Hint:

Angle at circumference = ½ × central angle.

Tangent Facts

https://cdn.mathpix.com/cropped/2025_07_01_a141ee9b5a6f03e0cd28g-14.jpg?height=794&width=1382&top_left_y=331&top_left_x=98

Key Tangent Rules

Recall these two key tangent properties to solve circle problems.

Key Points:

  • Tangent & Radius: They meet at \(90^{\circ}\).
  • Equal Tangents: From the same external point, their lengths are equal.

Circle Key Ideas

Centre, radius and diameter define a circle.

Chord joins two points; arc is curved path between them.

Arc length: \(L = 2\pi r \times \frac{\theta}{360^\circ}\).

Angle at centre equals twice angle at circumference.

Tangent meets circle once, ⟂ radius; equal tangents from one point.

Thank You!

Recap complete—you can now state every key idea with confidence.