CBSE Class 10 Mathematics (Standard) Chapter 10 Circles Important Questions 2027
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33 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 4.7 questions per paper (OR alternatives counted separately)
- 6 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 2023Board 2024Board 2025Sample 2026 | ★★☆☆☆ |
| 3 | Tangent perpendicular to radius at point of contact | Board 2024Board 2025Board 2026Sample 2026 | ★☆☆☆☆ |
| 4 | Lengths of tangents from an external point are equal (and proof) | Board 2026Sample 2025 | ★☆☆☆☆ |
Repeated questions from Circles
1. Prove equal tangents theorem
2×Board 2026Sample 20253/5 marks
Prove that the lengths of tangents drawn from an external point to a circle are equal. Using above result, find the length BC of triangle ABC. Given that, a circle is inscribed in triangle ABC touching the sides AB, BC and CA at R, P and Q respectively and AB = 10 cm, AQ = 7 cm, CQ = 5 cm.
2. Tangent between two parallel tangents subtends 90 at centre
2×Board 2024Sample 20263 marks
In Figure, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X'Y' at B. Prove that angle AOB = 90 degrees. [For Visually Impaired candidates: Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that angle APB = 2(angle OAB)]
3. Tangent-chord angle vs central angle MCQ
2×Board 2023Sample 20251 marks
In the given figure, a tangent has been drawn at a point P on the circle centred at O. If angle TPQ = 110 deg then angle POQ is equal to A) 110 deg B) 70 deg C) 140 deg D) 55 deg
4. Prove angle PTQ = 2 angle OPQ
2×Board 2023Board 20263 marks
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 angle OPQ.
5. Tangents at ends of a diameter are parallel
2×Board 2024Board 20251 marks
The tangents drawn at the extremities of the diameter of a circle are always : (A) parallel (B) perpendicular (C) equal (D) intersecting
6. Incircle in quadrilateral with one right angle
2×Board 2023Board 20243 marks
A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If AD is perpendicular to DC, BC = 30 cm and BS = 24 cm, then find the length DC.
Circles questions by paper
| Paper | Questions | Marks |
|---|---|---|
| Board 2022 | 2 | 6 |
| Board 2023 | 7 | 13 |
| Board 2024 | 7 | 13 |
| Board 2025 | 5 | 10 |
| Sample 2025 | 3 | 7 |
| Board 2026 | 5 | 10 |
| Sample 2026 | 4 | 7 |
Counts include both options of internal-choice (OR) questions.
Previous year questions: Circles
Board 2026 5 questions
In the given figure, PA and PB are tangents to a circle centred at O. If angle OAB = 15 degrees, then angle APB equals : (A) 30 degrees (B) 15 degrees (C) 45 degrees (D) 10 degrees
In the given figure, PA and PB are tangents to a circle centred at O. If angle AOB = 130 degrees, then angle APB is equal to : (A) 130 degrees (B) 50 degrees (C) 120 degrees (D) 90 degrees
In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR is cyclic.
Prove that the lengths of tangents drawn from an external point to a circle are equal.
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 angle OPQ.
Sample 2026 4 questions
A quadrilateral ABCD is drawn to circumscribe a circle. If BC = 7cm, CD = 4cm and AD = 3cm, then the length of AB is (A) 3cm (B) 4cm (C) 6cm (D) 7cm
The length of a tangent drawn to a circle of radius 9 cm from a point at a distance of 41cm from the centre of the circle is (A) 40cm (B) 9cm (C) 41cm (D) 50cm
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm and 8 cm respectively. Find the lengths of the sides AB and AC, if it is given that ar(triangle ABC) = 90 cm2. [For Visually Impaired candidates: A circle is inscribed in a right-angled triangle ABC, right angled at B. If BC = 7cm and AB = 24cm, find the radius of the circle]
In Figure, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X'Y' at B. Prove that angle AOB = 90 degrees. [For Visually Impaired candidates: Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that angle APB = 2(angle OAB)]
Board 2025 5 questions
The tangents drawn at the extremities of the diameter of a circle are always : (A) parallel (B) perpendicular (C) equal (D) intersecting
In the given figure, PA is a tangent from an external point P to a circle with centre O. If angle POB = 115 deg, then angle APO is equal to : (A) 25 deg (B) 65 deg (C) 90 deg (D) 35 deg
A person is standing at P outside a circular ground at a distance of 26 m from the centre of the ground. He found that his distances from the points A and B on the ground are 10 m (PA and PB are tangents to the circle). Find the radius of the circular ground.
In the given figure, O is the centre of the circle and BCD is tangent to it at C. Prove that angle BAC + angle ACD = 90 deg. [figure: circle centre O, chord CA, line BCD tangent at C]
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Sample 2025 3 questions
Which of the following statements is not true? A) A number of secants can be drawn at any point on the circle. B) Only one tangent can be drawn at any point on a circle. C) A chord is a line segment joining two points on the circle D) From a point inside a circle only two tangents can be drawn.
In the given figure, a tangent has been drawn at a point P on the circle centred at O. If angle TPQ = 110 deg then angle POQ is equal to A) 110 deg B) 70 deg C) 140 deg D) 55 deg
Prove that the lengths of tangents drawn from an external point to a circle are equal. Using above result, find the length BC of triangle ABC. Given that, a circle is inscribed in triangle ABC touching the sides AB, BC and CA at R, P and Q respectively and AB = 10 cm, AQ = 7 cm, CQ = 5 cm.
Board 2024 7 questions
Assertion (A): The tangents drawn at the end points of a diameter of a circle, are parallel. Reason (R): Diameter of a circle is the longest chord. Choose the correct option: (a) Both A and R are true and R is correct explanation of A. (b) Both A and R are true but R is not correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.
[case context: Right triangle backyard ABC, right angle at B, AB = 7 m, BC = 15 m; circular pit inside touches AC, BC, AB at P, Q, R with AP = x m.] Find the length of AR in terms of x.
[case context: Right triangle backyard ABC, right angle at B, AB = 7 m, BC = 15 m; circular pit inside touches AC, BC, AB at P, Q, R with AP = x m.] Write the type of quadrilateral BQOR.
[case context: Right triangle backyard ABC, right angle at B, AB = 7 m, BC = 15 m; circular pit inside touches AC, BC, AB at P, Q, R with AP = x m.] Find the length PC in terms of x and hence find the value of x.
[case context: Right triangle backyard ABC, right angle at B, AB = 7 m, BC = 15 m; circular pit inside touches AC, BC, AB at P, Q, R with AP = x m.] Find x and hence find the radius r of circle.
In the given figure, AB is a diameter of the circle with centre O. AQ, BP and PQ are tangents to the circle. Prove that angle POQ = 90 degrees.
A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If AD is perpendicular to DC, BC = 30 cm and BS = 24 cm, then find the length DC.
Board 2023 7 questions
In the given figure, PQ is tangent to the circle centred at O. If angle AOB = 95 degrees, then the measure of angle ABQ will be (A) 47.5 deg (B) 42.5 deg (C) 85 deg (D) 95 deg
[case context: discus thrower - discus travels along tangent AB to a circle of radius 75 cm with centre O, angle ABO = 30 degrees, PQ parallel to OA] Find the length of AB.
[case context: discus thrower - discus travels along tangent AB to a circle of radius 75 cm with centre O, angle ABO = 30 degrees, PQ parallel to OA] Find the length of OB.
[case context: discus thrower - discus travels along tangent AB to a circle of radius 75 cm with centre O, angle ABO = 30 degrees, PQ parallel to OA; P on AB, Q on OB] Find the length of AP.
[case context: discus thrower - discus travels along tangent AB to a circle of radius 75 cm with centre O, angle ABO = 30 degrees, PQ parallel to OA; P on AB, Q on OB] Find the length of PQ.
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 angle OPQ.
In the given figure, a circle is inscribed in a quadrilateral ABCD in which angle B = 90 degrees. If AD = 17 cm, AB = 20 cm and DS = 3 cm, then find the radius of the circle. [Figure: circle with centre O inscribed in quadrilateral ABCD touching sides at P, Q, R, S]
Board 2022 2 questions
Two concentric circles are of radii 4 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
In Figure 1, a triangle ABC with angle B = 90 degrees is shown. Taking AB as diameter, a circle has been drawn intersecting AC at point P. Prove that the tangent drawn at point P bisects BC.
Frequently asked questions
What are the most important topics in CBSE Class 10 Mathematics (Standard) 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 (Standard) exam?
Circles is part of the Geometry unit, which carries 15 marks in the official syllabus. In the analysed papers it had about 4.7 questions per paper.
Do questions from Circles repeat in board exams?
Yes — 6 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.