Circle: First Look

Circle

A circle is the set of all points in a plane that are equidistant (radius) from a fixed centre.

When defining a circle, always mention the plane, the radius, and the centre.

Arc & Chord

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

Arc AB (curved) and chord AB (straight)

How to tell them apart

An arc is the curved boundary between two points on a circle, while a chord is the straight line joining the same points.

Spotting this difference helps you apply circle theorems confidently.

Key Points:

  • Arc – curved part of the circumference.
  • Chord – straight segment connecting the arc’s endpoints.
  • Diameter is the longest chord; not every chord is a diameter.

Arc Length Formula

\[L = 2\pi r\,\frac{\theta}{360^{\circ}}\]

Bigger central angle means a longer arc. Use the formula to find \(L\) when \(r\) and \(\theta\) are known.

Variable Definitions

L arc length
r radius
θ central angle (degrees)

Applications

Finding the curved edge of tracks

Measure rail or running track length accurately.

Designing roundabouts

Determine curb length from roadway angles.

Major vs Minor Sector

https://cdn.mathpix.com/cropped/2025_07_01_a141ee9b5a6f03e0cd28g-05.jpg?height=824&width=906&top_left_y=343&top_left_x=874

What is a Sector?

A sector is the region enclosed by two radii and the arc between them.

It looks like a slice of pizza and is named by its central angle.

Key Points:

  • Minor sector – smaller region, central angle < 180°.
  • Major sector – larger region, central angle > 180°.
  • Major + minor sectors together form the full circle (360°).

Segment of a Circle

Segment of a Circle diagram

What is a Segment?

The region cut from a circle by a chord is called a segment.

Key Points:

  • Curved boundary → arc
  • Straight boundary → chord

Name the Parts!

Drag each term onto its matching description to reinforce circle vocabulary.

Draggable Items

Radius
Diameter
Chord
Arc
Sector
Segment

Drop Zones

Straight line through centre

Line from centre to edge

Curved part of circumference

Straight line joining two points on circle

‘Pizza slice’ region

Shaded region between chord and arc

Tip:

Drop all terms, then hit ‘Check’ to see your score!

Angle at Centre

https://cdn.mathpix.com/cropped/2025_07_01_a141ee9b5a6f03e0cd28g-09.jpg?height=671&width=658&top_left_y=413&top_left_x=535

Angle at Centre Theorem

The angle subtended at the centre of a circle by a chord is twice the angle subtended at any point on the circumference by the same chord.

Key Point:

  • \( \angle AOB = 2 \times \angle ACB \) (chord AB).

Quick Check – Angle Theorem

Question

If the angle at the circumference is 30°, what is the angle at the centre?

1
30°
2
60°
3
90°
4
120°

Hint:

Remember: centre angle = 2 × circumference angle.

Key Takeaways

A circle is the set of points fixed distance from its centre.

Arc is a curved edge; chord is a straight line joining two points.

Arc length connects radius, central angle, and π.

Sectors and segments slice the circle by radii or chords.

Angle at centre equals twice the angle on the circumference.

Next Steps

Practice problems and measure real-world circles to strengthen these ideas.

Thank You!

Now you can summarise circle parts and their angle relationships.