CIRCLES • TANGENTS • GEOMETRY PRACTICE
Circles Worksheet
with Online Quiz
See the right angle. Spot the equal tangents. Turn each diagram into a clear calculation or convincing proof.
Discover the Tangent
Explore contact points, perpendicular radii and point positions.
Touch and explore →Connect Equal Lengths
Use tangent pairs to solve lengths, algebra and perimeter problems.
Try the activities →Build Your Reasoning
Link angles, right triangles and theorem-based explanations.
Explore the challenges →Circles Online Quiz
Question 1 of 20 • Easy • Answered 0/20
How many common points does a tangent line have with a circle?
How the quiz works
Try 10 easy, five average and five HOTS questions. The 20-minute timer begins with your first answer or navigation. Selecting an answer opens the next question. Revisit questions using Previous or the numbered buttons. Each correct answer earns one mark with no negative marking. Solutions appear after submission or timeout.
Interactive Circles Games
Touch the Geometry
Tap a point or segment on the diagram, or use the matching button.
Select the centre of the circle.
Choose and play
Use the raised game selector to choose a diagram game, moving-point lab, theorem matching, statement sorting or proof builder. Each game contains five tasks.
Circle Theorem Activities
Answer entry help
Enter numbers without units. Fractions and radicals such as 7/2, 5√3 and 5*sqrt(3) are supported. For non-terminating decimals, use at least three decimal places. Your entries remain available while the page stays open.
Extra Circles Practice
Concentric Circle Explorer
The larger circle’s chord is tangent to the smaller circle.
Concentric radii are 17 cm and 8 cm. Find the full chord of the larger circle that touches the smaller.
Enter exact fractions or radicals, such as 5√3 or 5*sqrt(3). For decimals, use at least three decimal places.
About the extra sets
Explore concentric circles, triangle contact lengths, a garden case study, HOTS reasoning and creating valid models. Each of the five sets contains five tasks.
Circles Revision Tips for Students
- Locate the centre, point of contact and external point before calculating.
- Join the centre to a contact point to reveal a right angle.
- In △OAP, OP is the hypotenuse when PA is tangent at A.
- Use equal tangents only when they start at the same external point and touch the same circle.
- Use the non-reflex central angle in the relation ∠APB+∠AOB=180°.
- Label equal tangent pieces before adding sides or finding a perimeter.
- For a tangent chord between concentric circles, calculate the half-chord first.
- Give a reason for each proof step and check that all lengths are positive.
Teaching Tips for Parents and Teachers
- Use the moving-point game to compare interior, boundary and exterior points.
- Ask students to identify the hypotenuse before using Pythagoras.
- Colour tangent pairs from each external vertex differently.
- Compare a correct theorem statement with one missing its conditions.
- Ask learners to explain why a numerical example is not a general proof.
Circles Notes Formulas and Solved Questions
This Circles Class 10 worksheet focuses on tangent properties, lengths and geometrical reasoning. Use the online quiz to check the basics, then explore the clickable diagrams and written-answer activities. The worksheet download option on this page supports further notebook practice.
Tangent and Secant of a Circle
A tangent line has exactly one common point with a circle. That point is its point of contact. A secant has two distinct intersection points. From an interior point, no tangent is possible; from a point on the circle, one tangent is possible; from an external point, two are possible. A circle has infinitely many tangents in total because it has infinitely many points of contact.
Circles Class 10 Theorem 10.1
The tangent at a point of a circle is perpendicular to the radius through that point. If PA touches a circle with centre O at A, then OA ⟂ PA and ∠OAP=90°.
To understand the proof, suppose OA is not perpendicular to the tangent. Drop a perpendicular OM to the tangent, with M different from A. Then OM<OA, so M lies inside the circle. A line through an interior point must intersect the circle twice, contradicting that the line is a tangent. Therefore OA is perpendicular to it.
Circles Class 10 Theorem 10.2
The lengths of tangent segments from the same external point to the same circle are equal. For tangents PA and PB, join OA, OB and OP. The contact angles are right angles, OA=OB as radii, and OP is common. The right triangles OAP and OBP are congruent by RHS, so PA=PB.
Circles Class 10 Formula Sheet
Let r be the radius, d=OP the centre-to-external-point distance and t=PA the tangent length. For two distinct tangents PA and PB, A and B are the contact points.
| Result | Relationship |
|---|---|
| Radius at contact | OA ⟂ PA |
| Equal tangents | PA = PB |
| Right triangle | d² = r² + t² |
| Tangent length | t = √(d²−r²), with d>r |
| Missing radius | r = √(d²−t²) |
| Tangent and central angles | ∠APB = 180°−∠AOB, using the non-reflex angle AOB |
| Angle bisector | ∠APO = ∠OPB |
| Quadrilateral with an incircle | AB+CD = BC+DA |
Circles Class 10 Important Questions
Build revision around tangent definitions and counts, missing tangent or radius lengths, algebra using equal tangents, angles between tangents, theorem proofs and tangential polygon lengths. Concentric-circle chord questions combine a tangent radius with the perpendicular bisector of a chord. These are useful practice types rather than predictions of a future board paper.
Circles Class 10 Examples with Solutions
Find a tangent length
Radius OA=15 cm and external distance OP=25 cm. Find tangent PA.
PA² = OP²−OA²
PA² = 25²−15²
PA² = 625−225
PA² = 400
PA = 20 cm
Solve equal tangent expressions
PA=2x+9 and PB=5x−12 are tangents from P. Find x and the tangent length.
PA = PB
2x+9 = 5x−12
21 = 3x
x = 7
PA = 2(7)+9
PA = 23 cm
Find the angle between two tangents
The non-reflex central angle AOB is 132°. Find ∠APB.
∠APB+∠AOB = 180°
∠APB = 180°−132°
∠APB = 48°
The two contact angles in quadrilateral OAPB are each 90°.
Circles Class 10 Case-Based Questions with Solutions
Two paths to a circular fountain
A circular fountain has radius 10 m. Two tangent paths PA and PB begin at a point P that is 26 m from the centre O.
What is ∠OAP? It is 90° because a contact radius is perpendicular to a tangent.
Find one path length.
PA² = 26²−10²
PA² = 576
PA = 24 m
Find the combined path length.
PA+PB = 24+24
PA+PB = 48 m
Circles Class 10 Extra Questions with Answers
- OP=30 cm and radius=18 cm. Find the tangent length.
View solution
t² = 30²−18²
t² = 576
t = 24 cm
- PA=6x+5 and PB=4x+19. Find x.
View solution
6x+5 = 4x+19
2x = 14
x = 7
- The angle between two tangents is 74°. Find the non-reflex central angle between the contact radii.
View solution
∠AOB = 180°−74°
∠AOB = 106°
- A quadrilateral has an incircle. AB=14, BC=11 and CD=18 cm. Find DA.
View solution
AB+CD = BC+DA
14+18 = 11+DA
DA = 21 cm
- Concentric circles have radii 10 cm and 6 cm. Find the chord of the larger circle tangent to the smaller.
View solution
Half-chord = √(10²−6²)
Half-chord = 8 cm
Full chord = 16 cm
How to Solve NCERT Circles Exercises
For Exercise 10.1, revise the meaning of a tangent and point positions. For Exercise 10.2, practise joining radii, recognising right triangles, pairing equal tangents and writing proof reasons. Match question numbers to your own textbook edition. This page contains original practice and does not reproduce every NCERT exercise solution.
Current Syllabus and Official Sample-Paper Update
The official CBSE Mathematics syllabus for 2026–27 includes proofs of the perpendicular radius–tangent theorem and the equal-tangents theorem, with applications to problems. The current official Mathematics Standard sample paper includes a circle case study asking about a contact angle, tangent length and central angle. Practise connecting these skills. This is a verified sample-paper reference, not a claim about which exact questions repeat most often.
Common Circles Mistakes to Avoid
Do not equate tangents from unrelated external points, subtract r from OP to obtain a tangent length, place the right angle at the external point, or use a reflex central angle directly in the supplementary-angle formula. Distinguish Circles tangent questions from the separate Areas Related to Circles chapter.
Related Maths Worksheets and Revision
Circles Frequently Asked Questions
What is the difference between a tangent and a secant?
A tangent has one point in common with a circle. A secant meets it at two distinct points. A line missing the circle has no common point.
Why is a radius perpendicular to a tangent?
If the contact radius were not perpendicular, the perpendicular from the centre to the line would be shorter than the radius. Its foot would lie inside the circle, so the line would cross the circle twice. That contradicts the tangent condition.
How do I find the length of a tangent from an external point?
Join the centre O to the contact point A. Since OA is perpendicular to PA, use PA²=OP²−OA². Take the positive square root and include the unit in your written answer.
Why are two tangent lengths from the same external point equal?
The two contact radii are equal and perpendicular to their tangents. Joining the centre to the external point creates two right triangles with the same hypotenuse. They are congruent by RHS, giving equal tangent lengths.
How many tangents can be drawn from inside on and outside a circle?
From an interior point there are zero, from a point on the circle there is one, and from an exterior point there are two. A circle has infinitely many tangents overall.
How do I calculate the angle between two tangents?
Subtract the non-reflex angle between the contact radii from 180°. The two right contact angles account for the other 180° in the quadrilateral formed by the centre, contact points and external point.
Does the line joining the centre to the external point bisect the tangent angle?
Yes, when two distinct tangents are drawn to the same circle. The right triangles formed with the contact radii are congruent, so the two angles at the external point are equal.
How do I solve equal-tangent algebra questions?
Set the two tangent-length expressions equal, solve for the variable and substitute back. Check both lengths agree and are positive.
Which Circles questions are important for exam practice?
Cover theorem statements and proofs, tangent counts, missing lengths, algebra, angle calculations and tangential polygon problems. Use verified official papers for actual exam examples and practise reasons as well as calculations.
How do I find a chord tangent to a smaller concentric circle?
Join the common centre to the contact point. This radius is perpendicular to the chord and bisects it. With outer radius R and inner radius r, the full chord length is 2√(R²−r²).
What is the side-sum property of a tangential quadrilateral?
If a circle touches all four sides of ABCD, then AB+CD=BC+DA. Equal tangent pieces from each vertex prove the result. Checking this equation alone does not prove that an arbitrary drawn quadrilateral has an incircle.
Is Circles the same as Areas Related to Circles?
Circles focuses here on tangent geometry, lengths, angles and proofs. Areas Related to Circles studies measurements such as sectors and segments. Use the related worksheet link to practise that chapter separately.
Ready to Spot the Hidden Right Triangle?
Identify the contact point, join the radius and explain your next step.
Start the Circles Online Quiz
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