Class 10 · Chapter 10 · Geometry unit (15 of 80 marks)

Circles Class 10: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 54 board-style questions.

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54 questions
17 multiple choice, 6 assertion–reason, 12 very short answer, 11 short answer, 5 long answer, 3 case study
About 10 hours to master

Geometry unit: 15 of 80 theory marks (Triangles and Circles).

NCERT textbook: Chapter 10 on ncert.nic.in (PDF)

Revision notes

Tangent and secant

A line and a circle can have 0 common points (non-intersecting line), 1 common point (tangent) or 2 common points (secant).
The common point of a tangent and the circle is the point of contact. There is exactly one tangent at each point of a circle.
A tangent is a limiting case of a secant, when its two points of intersection coincide.
Number of tangents from a point: inside the circle, 0; on the circle, 1; outside the circle, 2.

Two key theorems (proofs can be asked)

Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Proof idea: Every point $Q$ of the tangent other than the contact point $P$ lies outside the circle, so $O Q > O P$ . Hence $O P$ is the shortest distance from $O$ to the line, and the shortest segment from a point to a line is the perpendicular.

Theorem 2. The lengths of the tangents drawn from an external point to a circle are equal.

Proof idea: In $△ O A P$ and $△ O B P$ : $O A = O B$ (radii), $O P$ is common, and $∠ O A P = ∠ O B P = 90^{\circ}$ . By RHS congruence $P A = P B$ . This also gives $∠ A P O = ∠ B P O$ : the centre lies on the bisector of the angle between the tangents.

Standard consequences

Tangent length: $P A = \sqrt{O P^{2} - r^{2}}$ .
Angle between tangents and angle at the centre are supplementary: $∠ A P B + ∠ A O B = 180^{\circ}$ , because $O A P B$ is a quadrilateral with two right angles.
Tangents at the ends of a diameter are parallel.
A quadrilateral $A B C D$ circumscribing a circle satisfies $A B + C D = B C + D A$ . So a parallelogram circumscribing a circle is a rhombus.
Incircle of a triangle with sides $a, b, c$ and semi-perimeter $s$ : the tangent lengths from $A, B, C$ are $s - a, s - b, s - c$ . For a right triangle with legs $a, b$ and hypotenuse $c$ , the inradius is $r = \frac{a + b - c}{2}$ .
In two concentric circles, a chord of the larger circle that touches the smaller one is bisected at the point of contact.

Worked example 1

The tangent from $P$ to a circle of radius 9 cm has length 12 cm. Find $O P$ . $O P = \sqrt{81 + 144} = 15$ cm.

Worked example 2

Tangents $P A, P B$ make $∠ A P B = 40^{\circ}$ . Find $∠ O A B$ .

$∠ A O B = 140^{\circ}$ . Triangle $O A B$ is isosceles, so $∠ O A B = \frac{180^{\circ} - 140^{\circ}}{2} = 20^{\circ}$ ( $= \frac{1}{2} ∠ A P B$ ).

Worked example 3

A circle touches the sides of quadrilateral $P Q R S$ with $P Q = 9$ , $Q R = 11$ , $R S = 8$ . Find $S P$ .

$P Q + R S = Q R + S P \Rightarrow S P = 9 + 8 - 11 = 6$ .

Common errors

Using $O P^{2} = P A^{2} - r^{2}$ : $O P$ is the hypotenuse, not a leg.
Claiming $∠ A P B = ∠ A O B$ . They are supplementary, not equal.
Quoting "tangents from an external point are equal" without naming the point. Write " $P A = P B$ (tangents from $P$ )".

Board-exam tips

Both theorem proofs are regularly asked for 3 marks. Learn them with a figure, the given, to prove, construction and proof.
In numerical problems, mark every pair of equal tangents on the figure first. The equations then write themselves.

Topics in this chapter: Tangent to a circle · Length of tangent · Tangents from an external point · Number of tangents from a point · Tangent perpendicular to radius.

Formulas for this chapter: Chapter 10 on the Class 10 formula sheet (free PDF and A3 poster).

Circles Class 10: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 7 pages · 594 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceLength of tangent

A circle has radius $8$ cm and a point $P$ is $17$ cm from its centre. The length of a tangent from $P$ to the circle is

Circle with centre O and radius 8 cm. A tangent from P touches the circle at T; P is 17 cm from O. The tangent length PT is marked with a question mark. — Radius 8 cm, OP = 17 cm

Show worked answer

The tangent is perpendicular to the radius: $\sqrt{17^{2} - 8^{2}} = \sqrt{225} = 15$ cm A1
Correct option: (a) $15$ cm

Q2·1 mark·Multiple choiceTangents from an external point

Tangents $P A$ and $P B$ from an external point $P$ touch a circle with centre $O$ at $A$ and $B$ . If $∠ A P B = 70^{\circ}$ , then $∠ A O B$ is

Q3·1 mark·Assertion–reasonTangent perpendicular to radius

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The tangents drawn at the two ends of a diameter of a circle are parallel.

Reason (R): A tangent to a circle is perpendicular to the radius through the point of contact.

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerTangents from an external point

From a point $P$ outside a circle with centre $O$ and radius $6$ cm, tangents $P A$ and $P B$ are drawn. If $∠ A O B = 120^{\circ}$ , find $∠ A P B$ and the length $P A$ .

Circle with centre O and radius 6 cm. Tangents PA and PB are drawn from the outside point P. Angle AOB is marked 120° and angle APB is marked with a question mark. — Tangents PA and PB; angle AOB = 120°

Show worked answer

Answer:

60^{\circ}

;

6 \sqrt{3}

$∠ O A P = ∠ O B P = 90^{\circ}$ , so $\begin{aligned} ∠ A P B & = 360^{\circ} - 180^{\circ} - 120^{\circ} \\ = 60^{\circ} \end{aligned}$ A1
$O P$ bisects $∠ A O B$ , so $∠ A O P = 60^{\circ}$ and $P A = O A \tan 60^{\circ} = 6 \sqrt{3}$ cm A1

Q5·2 marks·Short answerTangents from an external point

A quadrilateral $A B C D$ is drawn so that each of its sides touches a circle. If $A B = 7$ cm, $B C = 9$ cm and $C D = 8$ cm, find $A D$ .

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerTangents from an external point

Prove that the lengths of the two tangents drawn from an external point to a circle are equal.

Show worked answer

Answer: Proved

Let $P A$ and $P B$ be tangents from $P$ to a circle with centre $O$ . Join $O A$ , $O B$ and $O P$ M1
$∠ O A P = ∠ O B P = 90^{\circ}$ (tangent $⊥$ radius), $O A = O B$ (radii), $O P$ common, so $△ O A P ≅ △ O B P$ (RHS) A1
Hence $P A = P B$ (corresponding parts of congruent triangles) A1

Q7·3 marks·Short answerTangents from an external point

A circle is inscribed in $△ A B C$ , touching $A B$ , $B C$ and $C A$ at $P$ , $Q$ and $R$ . If $A B = 10$ cm, $B C = 12$ cm and $C A = 8$ cm, find $A P$ , $B Q$ and $C R$ .

Triangle ABC with a circle inside it touching AB at P, BC at Q and CA at R. AB is marked 10 cm, BC 12 cm and CA 8 cm. — Incircle of triangle ABC: AB = 10 cm, BC = 12 cm, CA = 8 cm

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerTangent perpendicular to radius

(a) Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. (b) $P T$ is a tangent at $T$ to a circle with centre $O$ and radius $9$ cm. If $P T = 12$ cm, find $O P$ and the shortest distance from $P$ to the circle.

Circle with centre O and radius 9 cm. PT is a tangent touching the circle at T, with PT marked 12 cm. O is joined to T and to P. — Radius 9 cm, tangent PT = 12 cm

Show worked answer

Answer:

O P = 15

cm;

6

(a) Let the tangent at $T$ be the line $X Y$ , and take any point $Q \neq T$ on $X Y$ . $Q$ lies outside the circle (a tangent meets the circle only at $T$ ) M1
So $O Q > O T$ for every such $Q$ : $O T$ is the shortest distance from $O$ to $X Y$ R1
The shortest segment from a point to a line is the perpendicular, so $O T ⊥ X Y$ A1
(b) $∠ O T P = 90^{\circ}$ : $O P = \sqrt{9^{2} + 12^{2}} = 15$ cm A1
Shortest distance from $P$ to the circle $= O P - r = 15 - 9 = 6$ cm A1

Q9·5 marks·Long answerTangents from an external point

$P A$ and $P B$ are tangents from a point $P$ to a circle with centre $O$ and radius $5$ cm, and $O P = 13$ cm. Find (a) the length $P A$ , (b) the area of the quadrilateral $O A P B$ , (c) the length of the chord $A B$ .

Circle with centre O and radius 5 cm. Tangents PA and PB touch it at A and B, and OP, dashed, is marked 13 cm. — Radius 5 cm, OP = 13 cm

Section E style · Case study (4 marks)

Q10·4 marks·Case studyTangents from an external point

Roads to a circular park. Two straight roads run from a junction $J$ and each just touches the circular boundary of a park, at $A$ and at $B$ . The park has centre $O$ and radius $30$ m, and $O J = 50$ m.

Find the length of road from $J$ to $A$ . [1 mark]
Explain why $J B$ has the same length as $J A$ . [1 mark]
Find the area of the region $O A J B$ enclosed by the two roads and the radii $O A$ and $O B$ . [2 marks]

A circular park with centre O and radius 30 m. Two straight roads from the junction J just touch the boundary at A and B; OJ, dashed, is marked 50 m. — Park of radius 30 m, OJ = 50 m

Show worked answer

$O A ⊥ J A$ : $J A = \sqrt{50^{2} - 30^{2}} = 40$ m A1
Tangents drawn from an external point to a circle are equal, so $J B = J A = 40$ m A1
$\begin{aligned} ar (O A J) & = \frac{1}{2} \times 30 \times 40 \\ = 600 m^{2} \end{aligned}$ M1
$\begin{aligned} ar (O A J B) & = 2 \times 600 \\ = 1200 m^{2} \end{aligned}$ A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 10 assertion–reason questions · the Class 10 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You know what a tangent is, how many tangents can be drawn from a point, and that a tangent is perpendicular to the radius.

Read first: Tangent and secant; Two key theorems (proofs can be asked) 8 practice questions · checkpoint: 4 questions, 5 marks, pass 80%

Practise step 1

Step 2

Board standard

You can use equal tangents from an external point and Pythagoras to find lengths and angles in board-style figures.

Read first: Two key theorems; Standard consequences; Worked examples 1-3 16 practice questions · checkpoint: 3 questions, 6 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write both theorem proofs and the long 'prove that' questions with a figure, given, to prove and reasons.

Read first: Two key theorems (proofs can be asked); Board-exam tips 7 practice questions · checkpoint: 3 questions, 14 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle incircles of triangles and quadrilaterals, tangents meeting chords, and the multi-step angle problems.

Read first: Standard consequences; Common errors 7 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 3 of the 54 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceTangent to a circle

The number of tangents to a circle that are parallel to a given secant is

(a)1
(b)2
(c)infinitely many
(d)0

Q2·1 mark·Multiple choiceTangent to a circle

The common point of a tangent and a circle is called the

(a)centre
(b)point of contact
(c)end of a secant
(d)chord point

Q3·1 mark·Assertion–reasonTangent to a circle

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): A straight line can touch a circle at two distinct points.

Reason (R): A secant intersects a circle at two distinct points.

(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.

Worked solution·2 marks·Very short answerTangents from an external point

The tangent from an external point $P$ to a circle of radius $r$ has length equal to $r$ . Find $O P$ in terms of $r$ , and the angle between the two tangents drawn from $P$ .

Full worked solution Free, no sign-in

$∠ O A P = 90^{\circ}$ : $O P = \sqrt{r^{2} + r^{2}} = r \sqrt{2}$ M1

$△ O A P$ is isosceles right-angled, so $∠ A P O = 45^{\circ}$ ; the angle between the tangents is $2 \times 45^{\circ} = 90^{\circ}$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original geometry problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I95 · Problem I102 · Problem I20 · Problem I88. All 28 →

Knowledge organiser: Circles on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Circles

Using OP² = PA² − r². Fix: OP is the hypotenuse, so OP² = PA² + r²; mark the right angle at the point of contact first.
Claiming ∠APB = ∠AOB. Fix: they are supplementary (∠APB + ∠AOB = 180°) because of the two right angles in quadrilateral OAPB.
Writing 'tangents are equal' without naming the point. Fix: write 'PA = PB (tangents from external point P)'; the reason carries the mark.
Theorem proofs without the given, to prove and a figure. Fix: learn both proofs (tangent ⟂ radius; equal tangents) in the full four-part format.
Circumscribed quadrilateral problems done without marking equal tangents. Fix: mark each pair of equal tangents on the figure with the same letter; AB + CD = AD + BC follows.
Using a secant property that is not in the syllabus. Fix: stick to the two theorems and Pythagoras; they are enough for every board question.

Exam strategy: how answers are step-marked · the proofs, written out · competency-based questions

Examiner Insights: common mistakes in CBSE Class 10 Maths, with fixes (our analysis of public sources) →

Circles in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 9, 23, 28, 29, 33
Sample paper 2 (Board standard): questions 10, 31, 33
Sample paper 3 (Above board): questions 11, 29, 33
Sample paper 4 (Above board): questions 10, 11, 19, 30
Sample paper 5 (95+ challenge): questions 11, 12, 29, 33

Want it against the clock? Take a timed 30-mark chapter test on Circles, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Circles: common questions

What is in CBSE Class 10 Maths Chapter 10, Circles?

The topics are: Tangent to a circle, Length of tangent, Tangents from an external point, Number of tangents from a point, Tangent perpendicular to radius. This page has revision notes for each, and 54 board-style practice questions with step-marked solutions.

How many marks is Circles worth?

It is in the Geometry unit, which carries 15 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Circles?

Using OP² = PA² − r². Fix: OP is the hypotenuse, so OP² = PA² + r²; mark the right angle at the point of contact first. Claiming ∠APB = ∠AOB. Fix: they are supplementary (∠APB + ∠AOB = 180°) because of the two right angles in quadrilateral OAPB.

Are the Circles important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Tangent and secant

Two key theorems (proofs can be asked)

Standard consequences

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Circles Class 10: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Circles

Circles in our sample papers

Circles: common questions

Work through the Route to 95 with your progress saved