C
CraftExam
medium difficulty5 min

Angle Subtended by Arc/Chord at Centre vs at Circumference

The angle subtended by an arc or chord at the centre of a circle is twice the angle subtended by it at any point on the remaining part of the circle.

In one line: Centre sees double.

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Understand it

If the same chord AB makes angle AOB at the centre and angle APB at the circumference on the same side arrangement, then the central angle is double the angle at the circumference. This result is used heavily in angle chasing.

Example

If chord AB subtends angle AOB = 100 degrees at centre O, then angle APB at the circumference on the major arc is 50 degrees.

Formula and rule

For same chord or arc AB, angle at centre = 2 x angle at circumference.

So angle AOB = 2 angle APB.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1Given angle APB = 35 degrees.
  2. 2Since angle AOB is at the centre for the same chord AB, angle AOB = 2 × 35 = 70 degrees.

Method you can reuse

  1. 1Find the chord or arc common to both angles.
  2. 2Decide which angle is at centre and which is at circumference.
  3. 3Double the circumference angle or halve the central angle.

Sketch it in your answer

Draw chord AB, centre O, point P on the circle, join OA, OB, PA and PB, and mark the central and circumference angles.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If angle AOB at the centre is 80 degrees for chord AB, what is angle APB at the circumference on the same arc relation?

Think first, then show answer
Angle APB is 40 degrees because the angle at the centre is twice the angle at the circumference for the same chord or arc.

Common wrong answer

If angle AOB = 120 degrees, writing angle APB = 240 degrees.

Why it loses marks: The circumference angle is half, not double.

Don't mix this up

Students often reverse the relation and write the angle at the circumference as twice the central angle.

Exam tip

Identify the same chord or same arc first. Then apply centre angle = 2 x circumference angle.

Angle Subtended by Arc/Chord at Centre vs at Circumference diagram

Open full screen to read every label clearly.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions may give either the central angle or circumference angle and ask for the other, often inside a multi-step circle diagram.
  1. 1

    Enough for 1 mark

    The angle subtended by an arc or chord at the centre of a circle is twice the angle subtended by it at any point on the remaining part of the circle.

  2. 2

    Add for 2 marks

    Example: If chord AB subtends angle AOB = 100 degrees at centre O, then angle APB at the circumference on the major arc is 50 degrees.

  3. 3

    Add for 3 marks

    This result is used heavily in angle chasing.

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