What is a Circle?

Circle

A circle is the set of all points in a plane that are at the same fixed distance, called the radius, from a fixed point, the centre.

Centre: fixed point. Radius: distance from centre to any point on the circle.

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

Curved arc and its straight chord

One curve, one straight line

Both share the same two boundary points, yet follow different paths.

Key Points:

  • Arc – curved part of the circumference between two points.
  • Chord – straight line segment joining the same two points.
  • Each chord slices off an arc; an arc is never straight.

Arc Length Formula

https://cdn.mathpix.com/cropped/2025_07_01_a141ee9b5a6f03e0cd28g-03.jpg?height=804&width=1431&top_left_y=333&top_left_x=578

Central angle θ marks the arc whose length we calculate.

Finding the Length of an Arc

An arc is a part of the circle’s circumference.

Compare its central angle \( \theta \) (in degrees) with the full circle \(360^\circ\) to compute length.

Key Points:

  • \( L = \frac{\theta}{360^\circ} \times 2\pi r \)
  • \(\frac{\theta}{360^\circ}\) is the fraction of the whole circle.
  • Measure \(r\) and \( \theta \), then substitute to find \(L\).

Sectors of a Circle

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

Minor sector shaded

What is a Sector?

A sector is the part of a circle enclosed by two radii and the arc between them.

Key Points:

  • Minor sector: central angle < 180°, the smaller slice.
  • Major sector: central angle > 180°, the larger slice.
  • Both sectors together complete the circle.

Segments Explained

https://cdn.mathpix.com/cropped/2025_07_01_a141ee9b5a6f03e0cd28g-06.jpg?height=726&width=729&top_left_y=382&top_left_x=503

Chord AB cuts the circle; the shaded area is the segment.

Segment of a Circle

A segment is the region of a circle bounded by a chord and the arc it subtends.

Shade the space between the chord and arc to visualise the segment clearly.

Key Points:

  • Chord = straight boundary.
  • Arc = curved boundary.
  • Minor segment is smaller; major segment is larger.

Area of Segment

\[ A = \tfrac{1}{2} r^{2} \left( \theta - \sin \theta \right) \]

Use when finding segment area. θ must be in radians.

Variable Definitions

A segment area
r radius of the circle
θ central angle (radians)

Applications

Exam Problems

Quickly find the shaded part between a chord and arc.

Design & Engineering

Calculate material removed by circular cut-outs.

Source: circle-geometry-extended-myp.pdf

Angle at Centre

Angle at centre theorem diagram

Angle-at-Centre Theorem

In any circle, the angle at the centre is twice the angle at the circumference that subtends the same arc.

Key Points:

  • Central angle = \(2 \times\) inscribed angle.
  • Both angles stand on the same arc.
  • Helps find unknown angles quickly.

Same Segment Angles

https://cdn.mathpix.com/cropped/2025_07_01_a141ee9b5a6f03e0cd28g-10.jpg?height=684&width=1606&top_left_y=386&top_left_x=527

Angles subtended by chord AB at C and D are equal.

Equal angles from a chord

A chord forms identical angles at any two points on the same segment of the circle.

Key Points:

  • Both angles lie on the same side of the chord (same segment).
  • No measurement tools needed—the theorem guarantees equality.

Angle in Semicircle

Right-angled triangle formed in a semicircle

Diameter AB joined to point C creates a right angle at C.

Why always 90°?

Connect the endpoints of a diameter to any point on the circle to form a triangle.

The angle opposite the diameter is always \(90^{\circ}\); this is called the angle in a semicircle.

Key Points:

  • Diameter subtends a semicircle of \(180^{\circ}\).
  • Angle in a semicircle is half of \(180^{\circ}\): \(90^{\circ}\).
  • Identify right angles quickly in circle problems.

Bisecting a Chord

https://cdn.mathpix.com/cropped/2025_07_01_a141ee9b5a6f03e0cd28g-12.jpg?height=705&width=709&top_left_y=389&top_left_x=472

Perpendicular from centre O meets chord AB at right angle.

Perpendicular Bisector of a Chord

From centre O, drop a 90° line to chord AB; it meets AB at M.

This line is the perpendicular bisector of AB.

Key Points:

  • OA = OB (radii).
  • Right angle creates congruent triangles OAM and OBM.
  • Congruent triangles give AM = MB, so the chord is bisected.

Multiple Choice Question

Question

A chord subtends an angle of 35° at the circumference. What is the angle at the centre?

1
35°
2
70°
3
140°
4
90°

Hint:

For any chord, the central angle is twice the angle at the circumference.

Match the Theorem

Drag each theorem name onto its matching statement to test your recall.

Draggable Items

Angle at Centre
Same Segment
Semicircle
Bisecting Chord

Drop Zones

Angle at the centre is twice the angle at the circumference.

Angles in the same segment are equal.

Angle in a semicircle is 90°.

A perpendicular from the centre to a chord bisects the chord.

Tip:

Picture the circle diagrams in your mind before dropping the item.

Circle Essentials Recap

Radius, chord and arc are the three basic parts of a circle.

Arc length: \( \ell = 2\pi r \tfrac{\theta}{360^\circ} \) (θ in degrees).

Sector area: \(A=\tfrac{\theta}{360^\circ}\pi r^{2}\); segment area: \(A=\tfrac{1}{2}r^{2}(\theta-\sin\theta)\).

Angle at centre theorem: central angle is twice the angle on the circumference.

Same-segment angles are equal; angle in a semicircle is 90°.

Perpendicular from centre bisects a chord; tangents from one point are equal and meet radius at 90°.

Thank You!

You can now summarise the key circle facts with confidence.