CBSE Class 10 Mathematics (Basic) Chapter 10 Circles Important Questions 2027
Last updated:
30 questions from Circles appeared in 7 of the 7 CBSE papers we analysed (2022–2026) — here are the topics that matter, the questions that repeat, and every previous-year question from this chapter.
Key facts
- Geometry unit — 15 marks in the official syllabus
- Asked in 7 of 7 analysed papers (2022–2026)
- About 4.3 questions per paper (OR alternatives counted separately)
- 7 question patterns repeated across papers
Important topics in Circles
| # | Topic | Asked in | Importance |
|---|---|---|---|
| 1 | Angle/length problems using tangent properties | Board 2022Board 2023Board 2024Board 2025Board 2026Sample 2025Sample 2026 | ★★★★☆ |
| 2 | Circle inscribed in triangle/quadrilateral problems | Board 2025Board 2026Sample 2025Sample 2026 | ★☆☆☆☆ |
| 3 | Tangent perpendicular to radius at point of contact | Board 2023Board 2024Sample 2025Sample 2026 | ★☆☆☆☆ |
| 4 | Lengths of tangents from an external point are equal (and proof) | Board 2023Sample 2025Sample 2026 | ★☆☆☆☆ |
Repeated questions from Circles
1. Central angle from angle between two tangents
3×Board 2022Board 2024Sample 20251/2 marks
In the given figure, PQ and PR are tangents to a circle centred at O. If ∠QPR = 35° then ∠QOR is equal to (A) 70° (B) 90° (C) 135° (D) 145°
2. Tangent length/radius via Pythagoras
2×Board 2023Sample 20261 marks
In the given figure, BC and BD are tangents to the circle with centre O and radius 9 cm. If OB = 15 cm, then the length (BC + BD) is : (a) 18 cm (b) 12 cm (c) 24 cm (d) 36 cm
3. Prove tangents from an external point are equal (theorem)
2×Board 2023Sample 20253 marks
Prove that the lengths of tangents drawn from an external point to a circle are equal.
4. Tangent length via right-triangle trig (given angle)
2×Board 2025Board 20261 marks
In the given figure, PA is a tangent to a circle with centre O. If OP = 10 cm, then the length of AP is : (A) 10√3 cm (B) 20 cm (C) 5 cm (D) 5√3 cm
5. Chord of larger concentric circle tangent to smaller
2×Board 2025Board 20262 marks
Two concentric circles are of radii 6 cm and 10 cm. Find the length of the chord of the larger circle which touches the smaller circle.
6. Rectangle circumscribing a circle is a square
2×Board 2025Board 20263 marks
Prove that a rectangle circumscribing a circle is a square.
7. Assertion-Reason on tangent properties
2×Board 2023Board 20241 marks
Assertion (A): If the PA and PB are tangents drawn to a circle with centre O from an external point P, then the quadrilateral OAPB is a cyclic quadrilateral. Reason (R): In a cyclic quadrilateral, opposite angles are equal. Options: (a) Both A and R are true and R explains A completely. (b) Both A and R are true but R does not explain A. (c) A is true but R is false. (d) A is false but R is true.
Circles questions by paper
| Paper | Questions | Marks |
|---|---|---|
| Board 2022 | 2 | 6 |
| Board 2023 | 5 | 7 |
| Board 2024 | 7 | 13 |
| Board 2025 | 3 | 6 |
| Sample 2025 | 5 | 10 |
| Board 2026 | 3 | 6 |
| Sample 2026 | 5 | 8 |
Counts include both options of internal-choice (OR) questions.
Previous year questions: Circles
Board 2026 3 questions
In the given figure, PA is a tangent to a circle with centre O. If OP = 10 cm, then the length of AP is : (A) 10√3 cm (B) 20 cm (C) 5 cm (D) 5√3 cm
Two concentric circles are of radii 6 cm and 10 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Prove that a rectangle circumscribing a circle is a square.
Sample 2026 5 questions
In the figure given below, radius r of the circle which touches the sides of the triangle is
A) 3 cm
B) 6 cm
C) 7 cm
D) 4 cm
From an external point Q, the length of tangent to a circle is 12 cm and the distance of Q from the centre of circle is 13 cm. The radius of circle (in cm) is
A) 10
B) 5
C) 12
D) 7
In the given figure, PA is a tangent from an external point P to a circle with centre O and diameter AB. If ∠POB = 115°, then measure of ∠APO is
A) 25°
B) 30°
C) 20°
D) 65°
Find the radius of the circle with centre at origin, if line l given by x + y = 5 is tangent to the circle at point P.
A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the given figure. Show that (AB + CD)/(AD + BC) = 1.
Board 2025 3 questions
In the given figure, PA is a tangent to a circle with centre O. If OP = 10 cm, then the length of AP is : (angle at P = 30°, OP = 10 cm) (A) 10√3 cm (B) 20 cm (C) 5 cm (D) 5√3 cm
Two concentric circles are of radii 6 cm and 10 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Prove that a rectangle circumscribing a circle is a square.
Sample 2025 5 questions
In the given figure, PQ and PR are tangents to a circle centred at O. If ∠QPR = 35° then ∠QOR is equal to (A) 70° (B) 90° (C) 135° (D) 145°
If perimeter of given triangle is 38 cm, then length AP is equal to (A) 19 cm (B) 5 cm (C) 10 cm (D) 8 cm
In two concentric circles, a chord of length 8 cm of the larger circle touches the smaller circle. If the radius of the larger circle is 5 cm, then find the radius of the smaller circle.
In the given figure, PA and PB are tangents to a circle centred at O. Prove that (i) OP bisects ∠APB (ii) OP is the right bisector of AB.
Prove that the lengths of tangents drawn from an external point to a circle are equal.
Board 2024 7 questions
Assertion (A): If the PA and PB are tangents drawn to a circle with centre O from an external point P, then the quadrilateral OAPB is a cyclic quadrilateral. Reason (R): In a cyclic quadrilateral, opposite angles are equal. Options: (a) Both A and R are true and R explains A completely. (b) Both A and R are true but R does not explain A. (c) A is true but R is false. (d) A is false but R is true.
[Case: Circular mirror as a circle with centre O; AP and AQ are tangents to the circle at P and Q with AP = 30 cm and ∠PAQ = 60°.] Find the length PQ.
[Case: Circular mirror as a circle with centre O; AP and AQ are tangents to the circle at P and Q with AP = 30 cm and ∠PAQ = 60°.] Find m∠POQ.
[Case: Circular mirror as a circle with centre O; AP and AQ are tangents to the circle at P and Q with AP = 30 cm and ∠PAQ = 60°.] Find the length OA.
[Case: Circular mirror as a circle with centre O; AP and AQ are tangents to the circle at P and Q with AP = 30 cm and ∠PAQ = 60°.] Find the radius of the mirror.
In two concentric circles, the radii are OA = r cm and OQ = 6 cm, as shown in the figure. Chord CD of larger circle is a tangent to smaller circle at Q. PA is tangent to larger circle. If PA = 16 cm and OP = 20 cm, find the length CD.
In given figure, two tangents PT and QT are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.
Board 2023 5 questions
The distance between two parallel tangents of a circle of diameter 7 cm is : (a) 7 cm (b) 14 cm (c) 7/2 cm (d) 28 cm
In the given figure, the perimeter of ΔABC is : (a) 30 cm (b) 15 cm (c) 45 cm (d) 60 cm
In the given figure, BC and BD are tangents to the circle with centre O and radius 9 cm. If OB = 15 cm, then the length (BC + BD) is : (a) 18 cm (b) 12 cm (c) 24 cm (d) 36 cm
Assertion (A) : A tangent to a circle is perpendicular to the radius through the point of contact. Reason (R) : The lengths of tangents drawn from the external point to a circle are equal. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A). (c) Assertion (A) is true, but Reason (R) is false. (d) Assertion (A) is false, but Reason (R) is true.
Prove that the tangents drawn from an external point to a circle are equal in length.
Board 2022 2 questions
In Figure 1, if tangents PA and PB drawn from a point P to a circle with centre O, are inclined to each other at an angle of 70 degrees, then find the measure of angle POA.
In Figure 3, the tangent l is parallel to the tangent m drawn at points A and B respectively to a circle centred at O. PQ is a tangent to the circle at R. Prove that angle POQ = 90 degrees.
Frequently asked questions
What are the most important topics in CBSE Class 10 Mathematics (Basic) Chapter 10 Circles?
By how often and how recently they were asked: Angle/length problems using tangent properties; Circle inscribed in triangle/quadrilateral problems; Tangent perpendicular to radius at point of contact.
How many marks does Circles carry in the CBSE Class 10 Mathematics (Basic) exam?
Circles is part of the Geometry unit, which carries 15 marks in the official syllabus. In the analysed papers it had about 4.3 questions per paper.
Do questions from Circles repeat in board exams?
Yes — 7 question patterns from this chapter appeared in two or more past board or official sample papers. They are listed on this page with the years they were asked.