How many practice questions are available for Circles?

SparkEd Math offers 60+ practice questions for Circles across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

Is Circles content free on SparkEd Math?

Yes! All Circles content on SparkEd Math is 100% free, including practice questions, visual solutions, lessons, and AI coaching. No credit card required.

How does SparkEd Math teach Circles differently from textbooks?

SparkEd Math uses visual step-by-step solutions for every Circles problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Circles content aligned with the CBSE syllabus?

Yes, all Circles questions and lessons on SparkEd Math are fully aligned with the Central Board of Secondary Education (CBSE) syllabus for Class 10. The content follows the official curriculum structure and exam pattern.

Circles — Class 10 CBSE (2025-26)

About Circles — Class 10 CBSE

Study tangent to a circle, number of tangents from a point, and related theorems. This topic is part of the CBSE Class 10 mathematics syllabus (chapter: Chapter 10). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Circles

Introduction to Circles and Tangents
Tangent Perpendicular to Radius (Theorem 10.1)
Length of Tangents from an External Point (Theorem 10.2)
Advanced Applications and Combining Theorems
Summary, Connections, and Practice

Interactive lesson · about 15 minutes · checkpoint question after every unit

Circles — solved examples for Class 10 CBSE

Example 1easy

Which of the following statements correctly defines a tangent to a circle?

A)A line that intersects a circle at two points.
B)A line that passes through the center of a circle.
C)A line that touches a circle at exactly one point.
D)A line segment that connects two points on a circle.

Step-by-step solution

A tangent is defined as a line that intersects the circle at exactly one point. This point is called the point of contact.
Options A describes a secant, option B describes a line containing a diameter, and option D describes a chord.

Answer: A line that touches a circle at exactly one point.

Example 2medium

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. The radius of the circle is:

A)7 cm
B)10 cm
C)12 cm
D)15 cm

Step-by-step solution

Let O be the centre of the circle and R be the point of contact of the tangent from Q. According to Theorem 10.1, the radius OR is perpendicular to the tangent QR.
Thus, triangle ORQ is a right-angled triangle with ∠ORQ = 90°. We can apply the Pythagorean theorem: OQ² = OR² + QR².
Given OQ = 25 cm and QR = 24 cm. Let OR = r (radius). So, 25² = r² + 24².
625 = r² + 576. This implies r² = 625 - 576 = 49. Therefore, r = √49 = 7 cm.

Answer: 7 cm

Example 3hard

A point P is located at a distance (x+1) cm from the center O of a circle with radius r cm. A tangent from P to the circle has a length of (x-1) cm. If the radius r is an integer, what is the value of r?

A)3 cm
B)4 cm
C)5 cm
D)6 cm

Step-by-step solution

Let the distance from P to O be PO = (x+1) cm, the tangent length be PT = (x-1) cm, and the radius be OT = r cm.
Since the tangent PT is perpendicular to the radius OT, triangle PTO is a right-angled triangle with hypotenuse PO.
By the Pythagorean theorem: PO² = PT² + OT².
Substitute the given values: (x+1)² = (x-1)² + r².
Expand and simplify: x² + 2x + 1 = x² - 2x + 1 + r² => 4x = r².
Since r is an integer, test the options. If r = 4, then 4x = 4² = 16, which means x = 4.
Check the lengths: PO = 4+1 = 5 cm, PT = 4-1 = 3 cm, OT = 4 cm. We verify 5² = 3² + 4² (25 = 9 + 16), which is true.

Answer: 4 cm

Practice questions on Circles

Q1.easy
From a point P lying *inside* a circle, how many tangents can be drawn to the circle?
1. A)Exactly one
2. B)Exactly two
3. C)Infinitely many
4. D)Zero
Show answer
Answer: Zero
Hint: Consider if a line starting from inside the circle can touch it at only one point without passing through.
Q2.easy
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the center O at a point Q so that OQ = 12 cm. What is the measure of ∠OPQ?
1. A)30°
2. B)60°
3. C)90°
4. D)180°
Show answer
Answer: 90°
Hint: Recall the relationship between the radius and the tangent at the point of contact.
Q3.easy
A student states that if a line AB is tangent to a circle at point P, and OP is a radius, then ∠OPA must be 60°. What is the error in this statement?
1. A)The angle should be 45°.
2. B)The angle should be 180°.
3. C)The angle should be 90°.
4. D)There is no error, the statement is correct.
Show answer
Answer: The angle should be 90°.
Hint: Remember the specific angle formed between a radius and a tangent at the point of contact.
Q4.medium
In the given figure (describe): A circle with centre O. PT and PQ are two tangents drawn from an external point P to the circle at T and Q respectively. If ∠TPQ = 60°, what is the measure of ∠TOQ?
1. A)60°
2. B)90°
3. C)120°
4. D)180°
Show answer
Answer: 120°
Hint: Consider the quadrilateral formed by O, T, P, Q. Remember the angle between radius and tangent, and the sum of angles in a quadrilateral.
Q5.medium
A point P is 13 cm away from the centre O of a circle. A tangent PT is drawn to the circle from P, and its length is 12 cm. What is the diameter of the circle?
1. A)5 cm
2. B)10 cm
3. C)13 cm
4. D)26 cm
Show answer
Answer: 10 cm
Hint: The tangent and the radius at the point of contact form a right angle. Use the Pythagorean theorem to find the radius first.
Q6.medium
If a quadrilateral ABCD is drawn to circumscribe a circle, then which of the following statements is always true?
1. A)AB + CD = AC + BD
2. B)AB + CD = BC + DA
3. C)AC + BD = BC + DA
4. D)AB + BC = CD + DA
Show answer
Answer: AB + CD = BC + DA
Hint: Recall Theorem 10.2 about the lengths of tangents drawn from an external point to a circle. Apply this theorem to each vertex of the quadrilateral.
Q7.hard
A quadrilateral ABCD is drawn to circumscribe a circle. If AB = 2y cm, BC = 3y cm, CD = (y+10) cm, and AD = (4y-10) cm, find the perimeter of the quadrilateral.
1. A)40 cm
2. B)45 cm
3. C)50 cm
4. D)55 cm
Show answer
Answer: 50 cm
Hint: Remember that for a quadrilateral circumscribing a circle, the sum of the lengths of opposite sides are equal. Use this property to first find the value of the variable.
Q8.hard
P is an external point from which two tangents PA and PB are drawn to a circle with centre O. If the angle between the tangents, ∠APB, is 70°, which of the following statements is TRUE?
1. A)Quadrilateral OAPB is a rhombus.
2. B)∠AOB = 110°.
3. C)The diagonals OP and AB bisect each other.
4. D)Triangle OAP is congruent to Triangle OBP, but not a right-angled triangle.
Show answer
Answer: ∠AOB = 110°
Hint: Recall that the radius is perpendicular to the tangent at the point of contact. Consider the sum of angles in the quadrilateral OAPB.
Q9.hard
Two circles with radii 9 cm and 4 cm touch each other externally. A common direct tangent touches them at points P and Q respectively. Find the length of the common tangent PQ.
1. A)10 cm
2. B)12 cm
3. C)13 cm
4. D)15 cm
Show answer
Answer: 12 cm
Hint: Draw a line segment from the center of the smaller circle perpendicular to the radius of the larger circle at the point of contact. This creates a right-angled triangle.

These are 9 of the 60 questions available for Circles. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.