⭕ CBSE · Class 10 · Mathematics · Chapter 10

Chapter 10
Circles

Complete chapter resources for CBSE Class 10 Maths — tangent theorems, chord properties, sample questions, previous year board questions, and instant AI question paper generation.

3Topics
6–8Board marks
8Sample questions
3PYQ included
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Key Concepts — Chapter 10
  • Tangent ⊥ radius: Tangent at point P ⊥ radius OP
  • Equal tangents: PA = PB (from external point P)
  • No. of tangents: 0 (inside), 1 (on), 2 (outside)
  • Angle in semicircle: Angle subtended = 90°
  • Tangent-chord angle: = angle in alternate segment
  • Length of tangent: PT = √(d² − r²), d = dist. to centre
Chapter Overview

What this chapter covers

Chapter 10 of NCERT Class 10 Mathematics extends the study of circles beyond basic definitions to a formal treatment of tangents. A tangent to a circle is a line that touches the circle at exactly one point — the point of tangency. The chapter opens by establishing how many tangents can be drawn to a circle from a given point: none when the point lies inside, exactly one when it lies on the circle, and exactly two when it lies outside.

Two fundamental theorems anchor the chapter. First, the tangent at any point of a circle is perpendicular to the radius drawn to the point of tangency. This result is used in nearly every proof and application question in board exams. Second, the lengths of the two tangents drawn from an external point to a circle are equal. This theorem drives a large family of numerical problems involving polygons circumscribed about a circle, where sides are expressed as sums of tangent lengths.

Board questions consistently test students' ability to write formal proofs of these two theorems, apply them to find unknown lengths in geometric figures (triangles, quadrilaterals circumscribing a circle), and combine them with Pythagoras' theorem for multi-step problems. Mastery of this chapter also supports Chapter 12 (Areas Related to Circles) and Chapter 13 (Surface Areas and Volumes).

Topics

What's inside Chapter 10

As per NCERT Class 10 Mathematics (CBSE syllabus)

Topic 1
Introduction to Tangents
Definitions of tangent and secant. Number of tangents from a point — zero (interior), one (on circle), two (exterior). Distinguishing tangent from chord and secant.
Topic 2
Tangent Perpendicular to Radius (Theorem 1)
Proof that the tangent at any point of a circle is perpendicular to the radius at that point. Converse: a line perpendicular to the radius at its outer end is a tangent. Applications using Pythagoras' theorem.
Topic 3
Equal Tangents from External Point (Theorem 2)
Proof that tangent lengths from an external point are equal. Applications: finding perimeter of circumscribed triangles and quadrilaterals, unknown side lengths, and angle relationships at the external point.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Ch 6 · Triangles
Pythagoras' theorem used in tangent-length problems
Class 9 Ch 10 · Circles
Chord properties, arc, angle subtended at centre
Chapter 10 Circles
Leads to
Ch 12 · Areas Related to Circles
Sector, segment areas — builds directly on circle geometry
Class 11 · Conic Sections
Tangent and normal to a circle using coordinate geometry
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 10 board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Objective 1 1 Number of tangents from a point, angle between tangents, tangent length
Very Short Answer 2 1 Find tangent length using Pythagoras, prove equal tangents in two steps
Short Answer / Proof 3 1 Formal proof of Theorem 1 or Theorem 2, or a perimeter problem
Long Answer / Application 4–5 0–1 Circumscribed polygon, multi-step tangent + angle problem
Total (approximate) 6–8 3–4 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 10 Maths Chapter 10. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
From an external point P, the number of tangents that can be drawn to a circle is: (a) 0 (b) 1 (c) 2 (d) Infinite
Q2 Easy 2 marks Short Answer
A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent drawn from P to the circle.
Q3 Medium 2 marks Short Answer
Two tangents PA and PB are drawn to a circle with centre O from an external point P. If ∠APB = 60°, find ∠AOB.
Q4 Medium 3 marks Proof
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of tangency.
Q5 Medium 3 marks Short Answer
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 80°, find ∠POA.
Q6 Hard 4 marks Proof
Prove that the lengths of the tangents drawn from an external point to a circle are equal. Using this result, if a quadrilateral ABCD is circumscribed about a circle, prove that AB + CD = AD + BC.
Q7 Hard 4 marks Word Problem
A circle is inscribed in a triangle ABC with AB = 10 cm, BC = 12 cm, and CA = 8 cm. Find the lengths of the tangents drawn from each vertex to the circle.
Q8 Hard 5 marks Case-Based
Two concentric circles of radii 5 cm and 3 cm have centre O. A chord AB of the larger circle is a tangent to the smaller circle at point P. (i) Prove that P is the midpoint of AB. (ii) Find the length of AB. (iii) Find the area of the trapezium formed by the two circles and the chord AB.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 10 Maths board papers — Circles chapter.

Board 20232 marks
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle. (All India 2023)
Board 20223 marks
Prove that the tangents drawn at the ends of a diameter of a circle are parallel. (Delhi 2022)
Board 20204 marks
In the figure, a triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC, and the area of triangle ABC. (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Circles carry in the CBSE Class 10 board exam? +
Typically 6–8 marks across 2–3 questions — one 1-mark MCQ on tangent properties, one 2-mark short answer on the tangent-radius theorem, and one 3–4 mark proof or application question. The chapter has featured in every CBSE Class 10 Maths board paper in recent years.
What are the two most important theorems from Circles for the CBSE board exam? +
The two theorems examiners test most often are: (1) The tangent at any point of a circle is perpendicular to the radius through the point of tangency — used in almost every proof-type question; and (2) The lengths of the two tangents drawn from an external point to a circle are equal — used in numerical and proof questions almost every year.
Do students need to write formal proofs in CBSE Class 10 Circles questions? +
Yes. The CBSE board regularly asks students to prove one of the two NCERT theorems (tangent perpendicular to radius, or equal tangent lengths from external point) in a 3-mark question. Students should practise writing these proofs step-by-step — stating the given, to prove, construction, and reason for each step — as partial credit is awarded for correct steps.
What is the difference between a tangent and a secant in Class 10 Circles? +
A tangent touches the circle at exactly one point (the point of tangency), while a secant is a line that intersects the circle at two distinct points. In Chapter 10, NCERT focuses entirely on tangents to a circle — their properties, the number of tangents from a point, and problems involving tangent lengths. Secants appear briefly in the introduction as a limiting case.
How do I generate a custom question paper for Circles using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 10 Mathematics, select Chapter 10 (Circles), set your preferred question-type mix (MCQ, proof, application) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.