Chapter 8 of 8

All 8 chapters
  1. 1Reflections and rotations, readunderstood
  2. 2Translations and combined moves, readunderstood
  3. 3Enlargements, readunderstood
  4. 4Congruent triangles, readunderstood
  5. 5Similar shapes, readunderstood
  6. 6Similar areas and volumes, readunderstood
  7. 7Arcs and sectors, readunderstood
  8. 8Angle rules in circles, readunderstood

M4.4.8 Chapter 8

Angle rules in circles

Two lines from the ends of an arc meet at the centre at 100°. What is the angle when they meet at the edge instead?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Use the rule that the angle at the centre is double
  • Find angles in a semicircle and in a cyclic quadrilateral
  • Use the rule that a tangent meets a radius at 90°

A chord is a straight line joining two points on a circle. The two points cut the circle into two arcs. A line from the centre to each end of a chord makes an angle at the centre. A line from a third point on the circle to each end makes an angle at the circumference. Both angles stand on the arc AB that does not contain C.

A circle with centre O and points A, B and C on it. Lines join O to A and to B, and C to A and to B. The angle at O between A and B is marked and labelled 150°. The angle at C between A and B is marked and labelled 75°. Both stand on the arc AB.75°150°ABCO
The angle at the centre is 150°. The angle at the circumference is 75°. They stand on the same arc AB.

The first rule. The angle at the centre is double the angle at the circumference on the same arc. Here 150° is double 75°. Measure with a protractor on a big drawing and you will see it holds for any circle.

Here is why. Draw the line from C through O to the other side of the circle, ending at D. The lines OA, OB and OC are all radii, so they are equal. That makes triangles OAC and OBC isosceles.

The same circle with a line from C through the centre O to D on the circle. At C the angle between A and D is labelled x, and the angle between D and B is labelled y. At O the angle between A and D is labelled 2x, and the angle between D and B is labelled 2y.xy2x2yABCDO
The line CD splits both angles into two parts. The angles x and y are at C. The angles 2x and 2y are at O.

Call the angle at C in triangle OAC x. Chapter 4 proved that the base angles of an isosceles triangle are equal, so the angle at A is also x. The angles on the straight line CD add to 180°, and so do the angles of triangle OAC. So the angle AOD is x + x = 2x. The same reasoning on the other side gives angle DOB = 2y. The angle at C is x + y and the angle at O is 2x + 2y. That is double. This drawing has O inside the angle at C. When O lies outside it, the same steps work with a subtraction in place of the addition.

The second rule follows at once. A diameter makes a straight angle of 180° at the centre. The angle at the circumference on the same arc is half of that, which is 90°. So the angle in a semicircle is 90°.

A circle with centre O. A diameter joins A and B. A third point C lies on the circle, and lines join C to A and to B. The angle at C is a right angle, marked with a small square and labelled 90°.90°ABCO
AB is a diameter. The angle at C is 90°, wherever C is on the circle.

The third rule is about a cyclic quadrilateral. It is a four sided shape with all four corners on a circle. Opposite angles add up to 180°. Why? The two angles at the centre on the arcs between the opposite corners add to 360°. Each angle at the edge is half of its centre angle. So the two edge angles add to half of 360°, which is 180°.

A circle with a four sided shape ABCD whose corners are on the circle. The angle at A is labelled 100°, at B 85°, at C 80° and at D 95°. The angles at A and C are opposite, and so are the angles at B and D.80°85°95°100°ABCD
Opposite corners: 100° + 80° = 180°, and 85° + 95° = 180°.

The fourth rule is about a tangent. A tangent is a straight line that touches a circle at one point. Every other point of the line lies outside the circle. So the radius to the touching point is the shortest line from the centre to the tangent. The shortest line to a line is the one at a right angle. So a tangent meets the radius at 90°.

A circle with centre O and a point P on it. A radius joins O to P. A straight line, the tangent, touches the circle at P and meets the radius at a right angle, marked with a small square and labelled 90°.tangent90°PO
The tangent touches the circle at P. The radius OP is at 90° to it.
A circle with centre O and points A, B and C on it. Lines join O to A and to B, and C to A and to B. The angle at O between A and B is labelled 140°. The angle at C between A and B is labelled x.x140°ABCO
Find the angle x.

Worked example

The angle at the centre of a circle is 140°. Find the angle x at the circumference, on the same arc.

  1. Both angles stand on the arc AB.
  2. The angle at the centre is double the angle at the circumference.
  3. So x = 140 ÷ 2 = 70°.

Answer: 70°

A circle with centre O. A diameter joins A and B. A point C lies on the circle. The angle at A between C and B is labelled 38°. The angle at C is marked as a right angle. The angle at B between A and C is labelled x.38°xABCO
Find the angle x.

Worked example

AB is a diameter and C is on the circle. Angle CAB is 38°. Find angle ABC.

  1. AB is a diameter, so the angle in a semicircle at C is 90°.
  2. The angles of triangle ABC add up to 180°.
  3. Angle ABC = 180 − 90 − 38 = 52°.

Answer: 52°

A circle with a four sided shape PQRS whose corners are on the circle. The angle at P is labelled 105° and the angle at Q is labelled 80°. The angle at R is labelled x and the angle at S is labelled y.x80°y105°PQRS
PQRS is a cyclic quadrilateral. Find x and y.

Worked example

PQRS is a cyclic quadrilateral. Angle P is 105° and angle Q is 80°. Find angle R and angle S.

  1. Opposite angles of a cyclic quadrilateral add up to 180°.
  2. Angle R is opposite angle P, so R = 180 − 105 = 75°.
  3. Angle S is opposite angle Q, so S = 180 − 80 = 100°.
  4. Check: the four angles are 105°, 80°, 75° and 100°. They add up to 360°.

Answer: R is 75° and S is 100°

A circle with centre O and a point P on it. T is a point outside the circle on the tangent at P. A radius joins O to P, and a line joins O to T. The angle at P between O and T is a right angle. The angle at O between P and T is labelled 50°. The angle at T is labelled x.x50°PTO
PT is a tangent. Find the angle x.

Worked example

PT is a tangent to a circle with centre O. Angle POT is 50°. Find angle OTP.

  1. The tangent PT meets the radius OP at 90°. So angle OPT = 90°.
  2. The angles of triangle OPT add up to 180°.
  3. Angle OTP = 180 − 90 − 50 = 40°.
  4. Check: 90° + 50° + 40° = 180°.

Answer: 40°

See it three ways

With things

Stand a coin up against the straight edge of a ruler. The ruler touches the coin at one point. Use the corner of a sheet of paper to check. The line from the middle of the coin to that point is square to the ruler.

As a picture

A circle with centre O and a point P on it. A radius joins O to P. A straight line, the tangent, touches the circle at P and meets the radius at a right angle, marked with a small square and labelled 90°.tangent90°PO

In symbols

The angle between a tangent and the radius to the touching point is 90°.

Try it

Check the angle in a semicircle

You need: a round tin lid or a compass, a ruler, the corner of a sheet of paper, a pencil.

  1. Draw round the tin lid, or use a compass. Mark the centre and draw a diameter through it.
  2. Mark any point C on the circle. Join C to both ends of the diameter.
  3. Fit the corner of a sheet of paper into the angle at C.
  4. Move C to a new place on the circle and try again.

What you will see: The corner fits exactly each time. The angle in a semicircle is 90° wherever C is.

If you cannot try it, this is what happens.
A circle with centre O. A diameter joins A and B. A point C lies on the circle and lines join C to A and to B. The angle at C is a right angle, marked with a small square and labelled 90°.90°ABCO

A wrong idea many people have

The angle at the centre equals the angle at the circumference on the same arc.

Why it feels right: Both angles are drawn from the same two points, so they look as if they should match.

The test that breaks it: Take a diameter AB. It makes a straight angle of 180° at the centre. The angle at any point C on the circle is 90°, as the tin lid test showed. They are not equal.

What is true: The angle at the centre is double the angle at the circumference on the same arc.

Key words

chord
a straight line joining two points on a circle.
stand on
an angle stands on the arc between the ends of its arms, on the side away from the angle.
semicircle
half of a circle, cut off by a diameter.
cyclic quadrilateral
a four sided shape with all four corners on a circle.
tangent
a straight line that touches a circle at one point.

Back to your guess

The answer is B: 50°, half of the angle at the centre.

The angle at the edge, on the same arc, is half the angle at the centre. So 100° at the centre makes 50° at the edge. This chapter shows why, and uses it to find angles.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The angle at the centre is 84°. What is the angle x at the circumference on the same arc?

    A circle with centre O and points A, B and C on it. Lines join O to A and to B, and C to A and to B. The angle at O between A and B is labelled 84°. The angle at C between A and B is labelled x.x84°ABCO
    Tap your answer.

    1. Not this one. The angle at the circumference is not the same as the angle at the centre. It is half as big. Go to M4.4.8. Try another.

    2. Yes, that is right. The angle at the centre is double the angle at the circumference on the same arc. So x = 84 ÷ 2 = 42°.

    3. Not this one. You doubled the angle. It is the angle at the centre that is double, so the angle at the circumference is half. Go to M4.4.8. Try another.

    See the answer

    42°

    The angle at the centre is double the angle at the circumference on the same arc. So x = 84 ÷ 2 = 42°.

    • If you chose A: The angle at the circumference is not the same as the angle at the centre. It is half as big. Go to M4.4.8.
    • If you chose C: You doubled the angle. It is the angle at the centre that is double, so the angle at the circumference is half. Go to M4.4.8.

  2. AB is a diameter and C is on the circle. Angle CAB is 25°. What is angle ABC?

    A circle with centre O. A diameter joins A and B. A point C lies on the circle. The angle at A between C and B is labelled 25°. The angle at B between A and C is labelled x.25°xABCO
    Tap your answer.

    1. Not this one. The angle at C is 90°, but the question asks for the angle at B. Use the angles of the triangle. Go to M4.4.8. Try another.

    2. Not this one. You took 25° away from 180°. The triangle ABC also has a right angle at C, which must be taken away too. Go to M4.4.8. Try another.

    3. Yes, that is right. The angle in a semicircle at C is 90°. The angles of the triangle add up to 180°, so angle ABC = 180 − 90 − 25 = 65°.

    4. Not this one. You copied the angle at A. The two angles are not equal, because triangle ABC is not isosceles. Go to M4.4.8. Try another.

    See the answer

    65°

    The angle in a semicircle at C is 90°. The angles of the triangle add up to 180°, so angle ABC = 180 − 90 − 25 = 65°.

    • If you chose A: The angle at C is 90°, but the question asks for the angle at B. Use the angles of the triangle. Go to M4.4.8.
    • If you chose B: You took 25° away from 180°. The triangle ABC also has a right angle at C, which must be taken away too. Go to M4.4.8.
    • If you chose D: You copied the angle at A. The two angles are not equal, because triangle ABC is not isosceles. Go to M4.4.8.

  3. ABCD is a cyclic quadrilateral. Angle A is 70°. What is angle C?

    A circle with a four sided shape ABCD whose corners are on the circle. The angle at A is labelled 70° and the opposite angle at C is labelled x.70°xABCD
    Tap your answer.

    1. Yes, that is right. Opposite angles of a cyclic quadrilateral add up to 180°. Angle C = 180 − 70 = 110°.

    2. Not this one. Opposite angles of a cyclic quadrilateral add up to 180°. They are not equal unless each one is 90°. Go to M4.4.8. Try another.

    3. Not this one. You took 70° away from 360°. That rule is for all four angles together. For two opposite angles the total is 180°. Go to M4.4.8. Try another.

    See the answer

    110°

    Opposite angles of a cyclic quadrilateral add up to 180°. Angle C = 180 − 70 = 110°.

    • If you chose B: Opposite angles of a cyclic quadrilateral add up to 180°. They are not equal unless each one is 90°. Go to M4.4.8.
    • If you chose C: You took 70° away from 360°. That rule is for all four angles together. For two opposite angles the total is 180°. Go to M4.4.8.

  4. PT is a tangent to a circle with centre O. Angle POT is 48°. What is angle OTP?

    A circle with centre O and a point P on it. T is a point outside the circle on the tangent at P. A radius joins O to P and a line joins O to T. The angle at P is a right angle. The angle at O between P and T is labelled 48°. The angle at T is labelled x.x48°PTO
    Tap your answer.

    1. Not this one. You took 48° away from 180° and forgot the right angle at P. The angles of triangle OPT include the 90°. Go to M4.4.8. Try another.

    2. Not this one. You copied the angle at O. The tangent makes a right angle with the radius, so the other angle is not equal. Go to M4.4.8. Try another.

    3. Yes, that is right. The tangent meets the radius at 90°, so angle OPT = 90°. The angles of triangle OPT add up to 180°, so angle OTP = 180 − 90 − 48 = 42°.

    See the answer

    42°

    The tangent meets the radius at 90°, so angle OPT = 90°. The angles of triangle OPT add up to 180°, so angle OTP = 180 − 90 − 48 = 42°.

    • If you chose A: You took 48° away from 180° and forgot the right angle at P. The angles of triangle OPT include the 90°. Go to M4.4.8.
    • If you chose B: You copied the angle at O. The tangent makes a right angle with the radius, so the other angle is not equal. Go to M4.4.8.

  5. Which statement about a circle is true?

    Tap your answer.

    1. Not this one. They are not equal. The angle at the centre is double the angle at the circumference on the same arc. Go to M4.4.8. Try another.

    2. Yes, that is right. This is the first rule of the chapter. The angle at the centre is double the angle at the circumference when both stand on the same arc.

    3. Not this one. Opposite angles add up to 180°. They are equal only when each is 90°. Go to M4.4.8. Try another.

    See the answer

    The angle at the centre is double the angle at the circumference on the same arc.

    This is the first rule of the chapter. The angle at the centre is double the angle at the circumference when both stand on the same arc.

    • If you chose A: They are not equal. The angle at the centre is double the angle at the circumference on the same arc. Go to M4.4.8.
    • If you chose C: Opposite angles add up to 180°. They are equal only when each is 90°. Go to M4.4.8.

  6. From an earlier chapter

    A sector has a radius of 8 cm and an angle of 45°. How long is its arc, in centimetres, to 1 decimal place?

    See the answer

    6.3

    The circumference is 2 × π × 8 = 50.3 cm. The sector is 45 out of 360 of it, so the arc is 6.3 cm.

  7. The angle at the centre of a circle is 120°. What is the angle at the circumference on the same arc, in degrees?

    See the answer

    60

    The angle at the circumference is half the angle at the centre: 120 ÷ 2 = 60°.

  8. ABCD is a cyclic quadrilateral. Angle B is 83°. What is angle D, in degrees?

    See the answer

    97

    Opposite angles of a cyclic quadrilateral add up to 180°. Angle D = 180 − 83 = 97°.

  9. A tangent touches a circle at P. The centre is O and T is a point on the tangent. Angle OTP is 31°. What is angle POT, in degrees?

    See the answer

    59

    The tangent meets the radius at 90°, so angle OPT = 90°. The angles of triangle OPT add up to 180°, so angle POT = 180 − 90 − 31 = 59°.

Keep it

Copy this into your exercise book.

Angle at the centre = 2 × angle at the circumference, same arc. Angle in a semicircle = 90°. Cyclic quadrilateral: opposite angles add to 180°. A tangent is square to the radius: 90°.

Draw: A circle with a chord and the angles at the centre and the edge. Add a tangent with a right angle mark.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.0 MB). It opens on any phone.