NCERT Class 10 Maths Chapter 10 Circles is the geometry chapter on tangents: what a tangent to a circle is, how it relates to the radius, and how many tangents can be drawn from a point. This page gives you the official chapter PDF from the NCERT source, a page-wise topic map taken from the textbook itself, both theorems with exam pointers, and an exercise-wise view of Exercises 10.1 and 10.2 — all from the textbook updated for the 2026-27 session.
Download the Official NCERT Class 10 Mathematics Chapter 10 PDF
The verified download for this chapter is in the official PDF table on this page. The link points to the file hosted on the official NCERT website, is checked before listing, and needs no sign-up. Save the chapter to your device once and study offline — the page numbers in the topic map below match the official PDF exactly, so you can jump straight to the section you need.
| Chapter | Official NCERT PDF |
|---|---|
| Chapter 10: Circles | Official NCERT PDF |
What You Study in Circles
The chapter has three teaching sections and a summary. Section 10.1 (Introduction) reconnects you with the circle vocabulary you already know and sets up the key question: what happens when a line and a circle meet? Section 10.2 (Tangent to a Circle) defines the tangent as a line that touches the circle at exactly one common point — the point of contact — and proves the chapter’s first theorem about the radius at that point. Section 10.3 (Number of Tangents from a Point on a Circle) examines how many tangents exist from a point inside, on, and outside the circle, and proves the second theorem about tangent lengths from an external point. Section 10.4 is the official three-point summary.
Three line-and-circle cases organise everything: a non-intersecting line touches the circle at no point, a secant intersects it at two distinct points, and a tangent touches it at exactly one point. Keep those three pictures in mind and every result in the chapter has a place.
Topic-Wise Index and Page Map of Circles
Section-wise index of Circles with the official NCERT page numbers (from the NCERT textbook):
| Section | Topic | Page | Type |
|---|---|---|---|
| 10.1 | Introduction | p. 1 | Topic |
| 10.2 | Tangent to a Circle | p. 2 | Topic |
| — | Exercise 10.1 | p. 4 | Exercise |
| 10.3 | Number of Tangents from a Point on a Circle | p. 4 | Topic |
| — | Exercise 10.2 | p. 8 | Exercise |
| 10.4 | Summary | p. 10 | Summary |
Key Definitions and Terms in Circles
- Circle — the collection of all points in a plane that are at a constant distance (the radius) from a fixed point (the centre).
- Chord — a line segment connecting any two points on a circle.
- Diameter — the longest chord of a circle; it passes through the centre.
- Secant — a line that intersects a circle at two distinct points.
- Tangent — a line that touches the circle at exactly one common point.
- Point of contact — the common point where a tangent touches the circle.
- Non-intersecting line — a line that does not intersect the circle at any point.
Important Formulas, Equations and Diagrams in Circles
Circles is a theorem chapter, not a formula chapter — your marks come from stating and applying two results exactly as the textbook words them.
- Theorem 10.1 — the tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Theorem 10.2 — the lengths of tangents drawn from an external point to a circle are equal.
Three companion results from the textbook’s remarks do steady exam work: the perpendicular drawn at the point of contact to the tangent passes through the centre; the two tangents from an external point subtend equal angles at the centre; and they are equally inclined to the segment joining the centre to that point. Because the radius meets the tangent at a right angle, the radius, the tangent length and the line from the centre to the external point form a right triangle — which is why many numerical questions in this chapter use the Pythagoras theorem. In diagrams, mark the right angle at the point of contact first; the figure then almost solves itself.
In-Text and Exercise Questions in Circles
The chapter carries worked examples and remarks between the theorems, then two exercises with distinct jobs:
- Exercise 10.1 (4 questions) — concept checks: how many tangents a circle can have, fill-in-the-blank vocabulary, one option-based numerical on tangent length, and one construction of a tangent and a secant parallel to a given line.
- Exercise 10.2 (13 questions) — the exam engine. It opens with option-based questions where you must justify the choice, then moves to proof and application work: tangent lengths from external points, circumscribed quadrilaterals, proving a parallelogram circumscribing a circle is a rhombus, and finding triangle sides from points of contact.
How to Prepare Circles for the Board Exam
- Read Sections 10.1 to 10.3 in the official PDF in order — the second theorem’s proof reuses the first.
- Learn both theorem statements word for word; write each proof once from memory.
- Finish Exercise 10.1 as a self-check on vocabulary and the tangent picture.
- Work Exercise 10.2 in order — justify the option questions, and in proofs always join the centre to the external point first.
- Close with the official summary in Section 10.4, and confirm the current unit-wise weightage from the CBSE syllabus for the 2026-27 session on the official CBSE website before you plan revision time.
Common Mistakes Students Make in Circles
These are the errors that actually cost marks in this chapter — most study pages will not tell you this plainly:
- Quoting Theorem 10.1 without the phrase “through the point of contact” — an incomplete statement is an incomplete answer in a proof question.
- Using equal tangent lengths in a proof without citing Theorem 10.2 as the reason for the step.
- Treating a chord or secant like a tangent in a figure — check how many points the line shares with the circle before writing anything.
- Applying the Pythagoras theorem without first justifying the right angle between the radius and the tangent.
- Memorising the two results but skipping the proof pattern; the proof questions in Exercise 10.2 reuse the same join-the-centre construction again and again.
How to Use NCERT Solutions for This Chapter
When you practise the chapter exercises, keep the official PDF open and check that every solved step follows the definitions, theorems, examples, or diagrams in the textbook. Do not memorise only the final answer; in proof and application questions, the reason written beside each step is what earns marks.
Official NCERT Summary of Circles
In this chapter, you have studied the following points :
1. The meaning of a tangent to a circle. 2. The tangent to a circle is perpendicular to the radius through the point of contact. 3. The lengths of the two tangents from an external point to a circle are equal.
Source: NCERT official textbook (Class 10 mathematics).
Frequently Asked Questions
What is a tangent in Class 10 Circles?
A line that touches the circle at exactly one common point (the point of contact); it is perpendicular to the radius through that point.
How many tangents can a circle have?
Infinitely many — one at each point of the circle — but at most two parallel tangents to a given line.
What are the theorems of Circles in Class 10?
Theorem 10.1: the tangent is perpendicular to the radius through the point of contact. Theorem 10.2: tangents from an external point have equal lengths.
Is there a formula for Circles in Chapter 10?
No — it is a theorem chapter; numericals combine the right angle at the point of contact with the Pythagoras theorem.