Circles, arcs, sectors and circle theorems
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Lesson notes
Parts of a circle
- The radius joins the centre to a point on the circle. A diameter is a chord through the centre and has length 2r.
- A chord is a segment joining two circle points. A tangent line meets the circle at exactly one point.
- An arc is part of the circumference. A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.
- For radius r, circumference = 2πr and area = πr². With θ in degrees, arc length = (θ/360) × 2πr and sector area = (θ/360) × πr².
A 60° sector of radius 6 cm
Circle theorems (the key ones)
- The angle at the centre is twice the angle at the circumference subtended by the same arc.
- Angles in the same segment are equal.
- The angle in a semicircle is 90°.
- Opposite angles of a cyclic quadrilateral add up to 180°.
- A tangent is perpendicular to the radius at the point of contact.
- Tangents from an external point to a circle are equal in length.
- The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
- A line from the centre perpendicular to a chord bisects the chord.
Angle subtended by a diameter
Giving reasons
- In angle problems write the angle and the theorem as a reason: "angle at centre is twice the angle at circumference".
- Mark equal radii: triangles with two radii are isosceles.
The radius and tangent at contact
- The radius to a tangent’s contact point is perpendicular to the tangent. In triangle OAT, O is the centre and A is the contact point, so angle OAT = 90°.
- Tangent segments from the same external point to the two contact points have equal lengths.
A radius meets a tangent at 90°
Slides
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Practice questions
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1.What is a chord?
Easy- Aa line from the centre to the edge
- Ba straight line joining two points on a circle
- Ca line touching a circle at one point
- Dpart of the circumference
2.What is a tangent?
Easy- Aa line that touches a circle at one point
- Ba line through the centre
- Ca slice of the circle
- Dpart of the circumference
3.What is the angle in a semicircle?
Easy- A45°
- B90°
- C180°
- D60°
4.What is the angle between a tangent and the radius drawn to its point of contact?
Easy- A45°
- B60°
- C90°
- D180°
5.The angle at the centre is 100°. What is the angle at the circumference on the same arc?
Easy- A50°
- B100°
- C200°
- D80°
6.Opposite angles of a cyclic quadrilateral: one is 110°. What is the opposite one?
Easy- A70°
- B110°
- C250°
- D90°
7.What is the arc length for a sector of angle 60° and radius 6 cm in terms of \π?
Easy- A2\π
- B\π
- C6\π
- D4\π
8.What is the area of a sector of angle 120° and radius 3 cm in terms of \π?
Easy- A3\π
- B\π
- C9\π
- D6\π
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