NCERT Solutions Class 10 Maths Chapter 10 helps you understand Circles through detailed solutions covering tangent properties, theorems, and constructions. You’ll learn how to prove that tangents from an external point are equal, find the length of tangents using the Pythagorean theorem, and solve problems involving circles touching each other. These 17 questions across 2 exercises build your geometry skills essential for board exams and competitive tests.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 10 Maths Chapter 10 Circles – Complete Guide
NCERT Class 10 Chapter 10 on Circles builds upon your foundational knowledge from Class 9 and introduces you to advanced concepts that carry significant weight in the CBSE board examination. You’ll explore tangents to circles, understand why a tangent is perpendicular to the radius at the point of contact, and learn the important theorem that tangents drawn from an external point are equal in length. This chapter typically contributes 5 marks to your board exam through a mix of 2-mark and 3-mark questions.
📊 CBSE Class 10 Maths Chapter 10 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 5 Marks |
| Unit Name | Geometry |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
You will work through practical applications of circle theorems, including problems involving tangent segments, secants, and the number of tangents that can be drawn from different positions relative to a circle. The chapter emphasizes logical reasoning and proof-based problems, helping you develop strong geometric thinking skills. You’ll encounter questions that require you to apply multiple concepts together, such as combining circle properties with triangle theorems and coordinate geometry.
Circles form the foundation for many real-world applications in engineering, architecture, and design. From understanding wheel mechanics to designing circular tracks and analyzing satellite orbits, the concepts you learn here have practical significance. The chapter also prepares you for advanced topics in Class 11 and competitive examinations like JEE and NEET.
Quick Facts – Class 10 Chapter 10
| 📖 Chapter Number | Chapter 10 |
| 📚 Chapter Name | Circles |
| ✏️ Total Exercises | 2 Exercises |
| ❓ Total Questions | 17 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
With focused practice of NCERT solutions and previous years’ CBSE question papers, you can confidently master this chapter and secure full marks in circle-based problems. The key is understanding the theorems deeply and practicing construction-based and numerical problems regularly to build speed and accuracy for your board exams.
NCERT Solutions Class 10 Maths Chapter 10 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 10.1 | Complete step-by-step solutions for 4 questions | 📥 Download PDF |
| Exercise 10.2 | Complete step-by-step solutions for 13 questions | 📥 Download PDF |
Circles – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Tangent-Radius Property \(\angle OPT = 90^\circ\) | The angle between a tangent and the radius at the point of contact is always 90 degrees. Note: O is the center of the circle, P is the point on the circle, and T is a point on the tangent. | Proving angles are right angles, solving problems involving right-angled triangles formed by tangents and radii. |
| Tangent Length from External Point \(PT = \sqrt{OP^2 – r^2}\) | Calculates the length of the tangent from an external point to a circle. Note: OP is the distance from the center of the circle to the external point, r is the radius of the circle, and PT is the length of the tangent. | Finding the length of a tangent when the distance from the center to the external point and the radius are known. |
| Equal Tangent Lengths \(PT_1 = PT_2\) | Tangents drawn from an external point to a circle are equal in length. Note: T1 and T2 are the points of tangency on the circle. | Proving tangent segments are equal, solving problems involving the perimeter of quadrilaterals circumscribing a circle. |
| Angle Bisector Property \(\angle OPT_1 = \angle OPT_2\) | The line joining the external point to the center of the circle bisects the angle between the tangents. Note: O is the center, P is the external point, T1 and T2 are points of tangency. | Finding angles formed by tangents from an external point, proving angle relationships. |
| Cyclic Quadrilateral Property \(\angle A + \angle C = 180^\circ\) | Opposite angles in a cyclic quadrilateral are supplementary (add up to 180 degrees). Note: A, B, C, and D are vertices of the quadrilateral on the circle. | Finding unknown angles in cyclic quadrilaterals, proving quadrilaterals are cyclic. |
| Angle Subtended at Center \(\angle AOB = 2\angle ACB\) | The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Note: O is the center of the circle, A and B are points on the circle. | Calculating angles when given the angle at the center or vice versa. |
| Angles in the Same Segment \(\angle ACB = \angle ADB\) | Angles in the same segment of a circle are equal. Note: A, B, C, and D are points on the circle such that C and D are in the same segment. | Proving angles are equal, finding unknown angles. |
| Tangent from External Point Theorem \(OA^2 = OT^2 + AT^2\) | Applies the Pythagorean theorem to relate the radius, tangent length, and distance from the external point to the circle’s center. Note: O is center, A is the external point, T is the point of tangency. | When needing to relate the radius, tangent length, and distance from the external point to the center of the circle. |
| Length of Common Tangent (Direct) \(L = \sqrt{d^2 – (r_1 – r_2)^2}\) | Length of the direct common tangent of two circles. Note: d is the distance between centers, r1 and r2 are the radii of the two circles. Applies when circles do not intersect. | Finding the length of a direct common tangent when radii and distance between centers are known. |
| Length of Common Tangent (Transverse) \(L = \sqrt{d^2 – (r_1 + r_2)^2}\) | Length of the transverse common tangent of two circles. Note: d is the distance between centers, r1 and r2 are the radii of the two circles. Applies when circles do not intersect. | Finding the length of a transverse common tangent when radii and distance between centers are known. |
Frequently Asked Questions – NCERT Class 10 Maths Chapter 10
📚 Related Study Materials – Class 10 Maths Resources
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|---|---|
| NCERT Class 10 Maths All Chapters | View Solutions |
| NCERT Class 10 Science Solutions | View Solutions |
| NCERT Class 10 Social Science | View Solutions |
| NCERT Class 10 English Solutions | View Solutions |