C
CraftExam
high importancemedium8 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.

Practice This Concept

Learn the concept

Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

    If angle AOB = 120 degrees, writing angle APB = 240 degrees is wrong because the circumference angle is half, not double.

Definition

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.

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.

Rule to remember

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

Memory hook

Centre sees double.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

If angle AOB = 120 degrees, writing angle APB = 240 degrees is wrong because the circumference angle is half, not double.

Exam tip

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

Quick check

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

Angle APB is 40 degrees because the angle at the centre is twice the angle at the circumference for the same chord or arc.

Study the angle subtended by arc/chord at centre vs at circumference diagram carefully

Use the labelled diagram to keep angle subtended by arc/chord at centre vs at circumference clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of angle subtended by arc/chord at centre vs at circumference in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on angle subtended by arc/chord at centre vs at circumference.

Revision cue

Revise angle subtended by arc/chord at centre vs at circumference through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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-mark answer

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. For same chord or arc AB, angle at centre = 2 x angle at circumference. So angle AOB = 2 angle APB. 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-mark answer

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. For same chord or arc AB, angle at centre = 2 x angle at circumference. So angle AOB = 2 angle APB. Given angle APB = 35 degrees. Since angle AOB is at the centre for the same chord AB, angle AOB = 2 x 35 = 70 degrees. Questions may give either the central angle or circumference angle and ask for the other, often inside a multi-step circle diagram. If angle AOB = 120 degrees, writing angle APB = 240 degrees is wrong because the circumference angle is half, not double.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept