Definition of Circle
Circle
A circle is the set of all points that are exactly the same distance (the radius) from one fixed point (the centre).
Find the centre first; any straight line from it to the curve is a radius.
Arc vs Chord
Circle diagram showing an arc (curved) and a chord (straight).
What’s the difference?
Both terms describe how the same two points on a circle are connected.
Knowing which is which lets you talk about circle parts accurately.
Key Points:
- Arc – curved slice of the circumference between two points.
- Chord – straight line segment joining the same two points inside the circle.
- Every diameter is a chord; it is simply the longest possible one.
Arc Length Formula
Central angle θ and radius r define the arc.
Formula (θ in degrees)
Arc length is the distance along the curved edge between two points on a circle.
Use \( \text{Arc Length} = 2\pi r \times \frac{\theta}{360^\circ} \) to calculate it when θ is measured in degrees.
Key Points:
- θ is the central angle in degrees.
- \( \frac{\theta}{360^\circ} \) gives the fraction of the full circle.
- Multiply this fraction by the circumference \( 2\pi r \).
Multiple Choice Question
Question
Self-check: Which statement is true about circles?
Hint:
Think of the chord that becomes the diameter when it passes through the centre.
Correct!
A chord through the centre is called the diameter, the longest chord of the circle.
Incorrect
Review circle facts: chords join two points on the circle, and the diameter is the special chord through the centre.
Sectors Explained
Circle showing a minor and major sector
Major vs Minor Sector
A sector is a slice of a circle bordered by two radii and their arc.
Knowing the size of the slice helps you spot which type it is.
Key Points:
- Smaller slice (central angle < 180°) → minor sector.
- Remaining part of the circle → major sector.
Segments Explained
Chord AB creates a minor segment.
What is a Segment?
A segment is the part of a circle cut off by a chord.
It looks like a slice or bite removed, bounded by the chord and the arc.
Key Points:
- Chord – straight line joining two points on the circle.
- Curved edge of the segment is an arc.
- Segments are named minor (small) or major (large).
Area of Segment
Circle segment illustration
Segment Area Formula (radians)
A segment is the part of a circle between a chord and its arc.
Use the formula below whenever the central angle is measured in radians.
Key Points:
- Area \(= \dfrac{1}{2} r^{2} (\theta - \sin \theta)\)
- \(\theta\) is the central angle in radians.
- Convert degrees to radians before applying the formula.
Multiple Choice Question
Question
θ = 270°. Which sector of the circle is larger?
Hint:
A minor sector spans less than 180°.
Correct!
Yes. 270° is more than half a circle, so it forms the major sector.
Incorrect
Check again: a minor sector is ≤ 180°, so 270° cannot be minor.
Alternate Segment
Angle between tangent and chord equals angle in opposite arc.
Alternate Segment Theorem
In circle theorems, this rule links a tangent–chord angle to an angle inside the circle.
Recognising this pair allows you to find unknown angles quickly.
Key Points:
- Angle between a tangent and its chord at the point of contact.
- Equals the angle the same chord subtends in the alternate (opposite) segment.
- Use it to spot equal angles when solving circle problems.
Angle at Centre
Central angle is twice the corresponding circumferential angle.
Centre Angle is Double
In circle geometry, the angle at the centre is special.
It is always twice the angle at the circumference on the same arc.
Key Points:
- \( \angle AOB = 2 \times \angle ACB \)
- Both angles subtend the same chord or arc AB.
- Use this circle theorem to find unknown angles quickly.
Same Segment Angles
Angles in the same segment are equal.
Circle Theorem
Circle theorem: Angles in the same segment are equal.
Any two angles subtended by the same chord on the same side share the same measure.
Key Points:
- Applies to any chord of a circle.
- Both angles lie on the circumference, same side of the chord.
- Handy for proving equal angles in geometric proofs.
Semicircle Right Angle
The angle in a semicircle is always 90°.
Diameter creates a right angle at the circumference
In any circle, a diameter subtends a right angle at every point on the circumference.
This circle theorem is often called the “angle in a semicircle” rule.
Key Points:
- Endpoints of the diameter act as a chord.
- Any point on the semicircle forms a 90° angle.
- Useful for spotting right triangles in circle problems.
Perpendicular Bisector
Circle Theorem: Perpendicular from Centre Bisects Chord
Draw radius \(OX\) so that \(OX \perp AB\). \(O\) is the centre of the circle.
Right-angled triangles \(\triangle OAX\) and \(\triangle OBX\) are congruent because \(OA = OB\) and \(OX\) is common.
Therefore \(AX = XB\). The perpendicular from the centre splits the chord equally, confirming the theorem.
Key Points:
- Line starts at circle centre \(O\).
- Meets chord at 90°.
- Forms two congruent right-angled triangles.
- Hence, chord is bisected: \(AX = XB\).
Cyclic Quadrilateral
Opposite angles A & C and B & D form straight lines.
Opposite Angles Are Supplementary
A quadrilateral with all four vertices on one circle is called cyclic.
Circle theorem: each pair of opposite interior angles sums to \(180^{\circ}\).
Key Points:
- All four vertices lie on the same circle.
- \(∠A + ∠C = 180^{\circ}\).
- \(∠B + ∠D = 180^{\circ}\).
Tangent Facts
Two Essential Tangent Properties
After this slide, you should be able to state both key tangent properties clearly.
Key Points:
- The tangent is perpendicular to the radius at the point of contact.
- Tangents drawn from the same external point are equal in length.
Tangent–Secant Rule
Tangent PA and secant PBC from point P
Power of a Point
From external point P, draw tangent PA and secant PBC to the circle.
Power of a Point states the square of the tangent equals the external part times the whole secant.
Key Points:
- Rule: \(PA^{2}=PB\cdot PC\)
- Use it to find any missing length when two are known.
- Valid for every circle sharing the same external point.
Secant–Secant Rule
Intersecting secants from point P
Power of a Point result
From an outside point P, draw two secants \(PAB\) and \(PCD\) to the circle.
The Power of a Point theorem states the products of their parts are equal.
Key Points:
- \(PA \cdot PB = PC \cdot PD\)
- Product of near and far parts of each secant is equal.
- Quickly find unknown lengths using the relation.