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Circles

Circles

Parts of a circle and the right-angle theorems

Properties Paper 1 + 2 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Parts of a circle
  4. Two right angles
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 4.1 and 4.7: the circle and its first two theorems
  • 4.1Use the words radius, diameter, chord, tangent, arc, sector and segment.
  • 4.7Use: the angle in a semicircle is 90°.
  • 4.7Use: the angle between a tangent and a radius is 90°, with triangle facts and Pythagoras.

Before you start: Angle facts. Next: More circle theorems.

Formula bank

Angle in a semicircleLearn itthe angle in a semicircle is 90°the triangle's longest side is a diameter.Use it when a diameter and a point on the circle.
Tangent and radiusLearn itthe angle between a tangent and a radius is 90°Use it when a tangent touching the circle.
Two radiiLearn ittwo radii make an isosceles trianglethe base angles are equal.Use it when the centre O joined to two points.

Parts of a circle

Parts of a circleOradiussegmentsectortangentchord
Chord, tangent, radius, sector and segment.
WordMeaning
radiuscentre to the circle
diameterchord through the centre (2 × radius)
chordstraight line joining two points on the circle
tangentstraight line touching the circle at one point
arcpart of the circumference
sectorregion between two radii and an arc
segmentregion between a chord and an arc

Try it yourself

1
Name the region between two radii and an arc.
Show answer

Answer sector

2
Name a straight line joining two points on a circle.
Show answer

Answer chord

Two right angles

Angle in a semicircleO34°ABC
AB is a diameter, so angle ACB = 90°.
Tangent meets the radius at 90°OTP
The radius OT meets the tangent at 90°.

General methodHow to use the two right-angle theorems

  1. Look for a DIAMETER with a third point on the circle: that corner is 90° (angle in a semicircle).
  2. Look for a TANGENT: the radius to the touching point meets it at 90°.
  3. Mark the right angle, then use angles in a triangle (180°) or Pythagoras.
  4. Look for two radii from the centre: an isosceles triangle.

Example PT is a tangent at T to a circle centre O. OT = 5 cm, PT = 12 cm. Find OP.

Right angle at T, so Pythagoras.

  1. \(OP^{2}=5^{2}+12^{2}=169\)

Answer OP = 13 cm

Try it yourself

1
AB is a diameter, C on the circle, angle CAB = 41°. Angle CBA?
Show answer

Answer 49

2
Radius 6 cm, tangent length 8 cm. Distance from P to the centre (cm)?
Show answer

Answer 10

Ways they test it

Type 1 Name the parts

"Complete the labels: chord, tangent, sector, segment." (2 marks)

Spot it a labelled diagram.

  1. Line or region?
  2. Two radii = sector; chord = segment.

Type 2 Angle in a semicircle

"AB is a diameter. Angle CAB = 34°. Find angle ABC, giving a reason." (2 marks)

Spot it a diameter and a triangle.

  1. Angle ACB = 90° (angle in a semicircle).
  2. 180 − 90 − 34.

For example angle ACB = 90°

    Answer 56°

    Type 3 Tangent + Pythagoras or trig

    "PT is a tangent. OT = 7 cm, angle OPT = 24°. Find OP." (3 marks)

    Spot it a tangent and a length.

    1. Right angle at T.
    2. Pythagoras or SOHCAHTOA.

    For example \(\sin24^\circ=\dfrac7{OP}\)

      Answer 17.2 cm

      Mistakes to avoid

      • ✗ Sector and segment swapped✓ sector: two radii; segment: a chordLearn the pictures.
      • ✗ Right angle at the centre✓ right angle at the circumferenceSemicircle theorem.
      • ✗ Missing the tangent right angle✓ OT ⟂ tangentIt is usually not marked.

      Summary

      • Sector: between two radii. Segment: between a chord and an arc.
      • Angle in a semicircle = 90°.
      • Tangent ⟂ radius at the point of contact.
      • Two radii → isosceles triangle.

      If you see …

      • a diameter in a trianglethe angle opposite it is 90°.
      • a tangentdraw the radius: right angle.

      Practice

      Practice: easy, medium, hard

      Easy

      1
      Name the region between a chord and an arc.
      Show answer

      Answer segment

      2
      AB is a diameter, C on the circle, angle CAB = 34°. Angle ABC?
      Show answer

      Answer 56

      3
      PT is a tangent at T. OT = 5, PT = 12. OP?
      Show answer

      Answer 13

      Medium

      1
      AB is a diameter of a circle centre O; C is on the circle; angle CAB = 27°. Find angle BOC.
      Guide me

      What is it asking? an angle

      What do we know? angle ABC = 63° (semicircle)

      Plan OB = OC: isosceles

      Show working
      1. angle OCB = 63°
      2. angle BOC = 180 − 126 = 54°

      Answer 54

      2
      Radius 8 cm; P is 17 cm from the centre. Tangent length PT (cm)?
      Guide me

      What is it asking? a length

      What do we know? right angle at T

      Plan \(17^{2}-8^{2}\)

      Show working
      1. \(\sqrt{225}=15\)

      Answer 15

      3
      PT is a tangent at T; angle TOP = 58°. Angle OPT?
      Guide me

      What is it asking? an angle

      What do we know? angle OTP = 90°

      Plan 180 − 90 − 58

      Show working
      1. 32°

      Answer 32

      Hard

      1
      PT is a tangent at T to a circle centre O. OT = 7 cm and angle OPT = 24°. Find OP to 1 d.p.
      Guide me

      What is it asking? a length

      What do we know? right angle at T

      Plan \(\sin24^\circ=\tfrac7{OP}\)

      Show working
      1. \(OP=7\div\sin24^\circ=17.2\)

      Answer 17.2

      PartAnswer / workingMarksGuidance
      90° at TB1
      \(\sin24^\circ=\tfrac7{OP}\)M1
      17.2A1
      Total: 3 marks
      2
      PT is a tangent at T. OT = 5 cm, PT = 12 cm. Find angle TOP to 1 d.p.
      Guide me

      What is it asking? an angle

      What do we know? right angle at T

      Plan \(\tan TOP=\tfrac{12}5\)

      Show working
      1. \(\tan^{-1}2.4=67.4^\circ\)

      Answer 67.4

      PartAnswer / workingMarksGuidance
      \(\tan^{-1}\tfrac{12}5\)M1
      67.4A1
      Total: 2 marks
      3
      AB is a diameter, C is on the circle, AC = 6 cm and BC = 8 cm. Find the radius.
      Guide me

      What is it asking? a radius

      What do we know? angle ACB = 90°

      Plan AB by Pythagoras, then halve

      Show working
      1. \(AB=\sqrt{36+64}=10\)
      2. radius 5

      Answer 5

      PartAnswer / workingMarksGuidance
      AB = 10M1
      5A1
      Total: 2 marks

      Can I do this?

      Anything unticked: revise that section first. All ticked: take the quiz.

      Quiz

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