Angle at the centre and in the same segment: Class 9 practice
The angle subtended by an arc at the centre is twice the angle it subtends at any point on the remaining circle. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Angle sum of triangle
- Isosceles triangles
The idea behind angle at the centre and in the same segment
The angle subtended by an arc at the centre is twice the angle it subtends at any point on the remaining circle. Angles in the same segment are equal, and the angle in a semicircle is 90°.
Step-by-step method
- Find the arc both angles stand on.
- Centre angle = 2 × circumference angle.
- Use 'same segment' or 'semicircle' facts when they apply.
Worked example 1 (Basic)
An arc subtends 65° at a point on the circle. What angle does it subtend at the centre?
- Centre angle = 2 × 65° = 130°
- Answer: 130°
Worked example 2 (Standard)
∠AOB = 52° at the centre O. Find ∠ACB for C on the major arc.
- ∠ACB = ½ × 52° = 26°
- Answer: 26°
Worked example 3 (Challenge)
O is the centre; OA = OB and ∠OAB = 23°. Find ∠ACB for C on the major arc.
- △OAB is isosceles: ∠AOB = 180° − 2(23°) = 134°
- ∠ACB = 67°
- Answer: 67°
Common mistakes and misconceptions
- Halving the wrong way
- Using the reflex angle by mistake
Practice questions
- (Basic) An arc subtends 78° at a point on the circle. What angle does it subtend at the centre?
- (Basic) An arc subtends 80° at a point on the circle. What angle does it subtend at the centre?
- (Standard) ∠AOB = 150° at the centre O. Find ∠ACB for C on the major arc.
- (Standard) ∠AOB = 94° at the centre O. Find ∠ACB for C on the major arc.
- (Challenge) O is the centre; OA = OB and ∠OAB = 47°. Find ∠ACB for C on the major arc.
- (Challenge) O is the centre; OA = OB and ∠OAB = 61°. Find ∠ACB for C on the major arc.
Answer key with full working
- Q1: Centre angle = 2 × 78° = 156° → Answer: 156°
- Q2: Centre angle = 2 × 80° = 160° → Answer: 160°
- Q3: ∠ACB = ½ × 150° = 75° → Answer: 75°
- Q4: ∠ACB = ½ × 94° = 47° → Answer: 47°
- Q5: △OAB is isosceles: ∠AOB = 180° − 2(47°) = 86° → ∠ACB = 43° → Answer: 43°
- Q6: △OAB is isosceles: ∠AOB = 180° − 2(61°) = 58° → ∠ACB = 29° → Answer: 29°
Syllabus note
This skill is listed under "Circles" for Class 9 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is angle at the centre and in the same segment in?
"Circles", Class 9 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.