CAT Circles Questions 2026: 15 Theorems + Solved Questions, PYQs & Free PDF

Quick Answer: CAT circles questions appear in the Quantitative Aptitude section and test tangents, chords, arcs, sectors, angle properties, cyclic quadrilaterals and circles in coordinate geometry. They are theorem-driven rather than calculation-heavy - almost every question turns on recognising which of about fifteen properties applies, after which the arithmetic is short.

Geometry contributes roughly 3 to 5 of the 22 Quantitative Aptitude questions, with circles typically accounting for one or two of those, and occasionally none as a standalone question.

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Key takeaways

• Circles questions are decided by theorem recognition, not calculation. Learn the fifteen properties and most become 90-second questions.
• Circles is typically one or two questions out of 68. Worth learning properly; not worth over-investing in.
• Draw the diagram before you read the options. More circles questions are lost to a cramped figure than to a wrong formula.
• Previous year questions recycle four configurations. Learn those and you have seen most of what CAT exam 2026 asks.

What Are CAT Circles Questions?

A circles question gives you a configuration - a circle with tangents, chords, inscribed angles, or a second circle - and asks for a length, an angle, an area or a ratio.

What separates them from the rest of CAT quant is that the difficulty sits almost entirely in recognition, not computation. Once you spot that two chords meet inside a circle, the intersecting chords theorem gives you the answer in one line.

Download CAT Circles Questions PDF with Answers

How Many Circle Questions Actually Appear in CAT?

Figures on this topic contradict each other badly. You will find claims that geometry is a third of the Quant section, that it carries 20% weightage, and that it runs 4 to 6 questions - all on pages competing for this keyword.

Here is what can actually be said. The IIMs publish no topic-wise breakdown of any section, so every figure - including the one below - is an estimate from candidate response sheets and published post-exam analyses.

Those analyses consistently indicate that geometry accounts for roughly 3 to 5 of the 22 Quantitative Aptitude questions. Within that, circles typically drive one or two, and in some years none as a standalone question.

Two consequences worth being clear about.

CAT Circles Questions is not a high-volume topic and you should not treat it like one. One or two questions out of 68 does not justify weeks of study.

But it is unusually cheap to learn. The theorem set is small and closed, and these properties keep appearing inside triangle, quadrilateral and coordinate questions well beyond the ones labelled "circles". That is the honest reason to study it - not that it wins three questions, but that a few hours removes a category you would otherwise skip. For the wider topic, see CAT geometry questions.

Practice CAT Circles Questions With Answers

1. Tangent Radius

A tangent to a circle is drawn at point P. If O is the center of the circle, then ∠OPT is:

1. 30°
2. 45°
3. 90°
4. 180°

Answer: C

2. Chord Distance

A circle has radius 13 cm. A chord is 10 cm long. The perpendicular distance of the chord from the center is:

1. 10 cm
2. 11 cm
3. 12 cm
4. 13 cm

Answer: C

Solution: Half chord = 5. Distance = √(13² − 5²) = √144 = 12.

3. Angle at Center

If an arc subtends 40° at the circumference, the angle subtended at the center is:

1. 20°
2. 40°
3. 60°
4. 80°

Answer: D

4. Equal Chords

In the same circle, equal chords subtend:

1. Equal angles at the center
2. Equal angles at the circumference only
3. Different central angles
4. None of these

Answer: A

5. Intersecting Chords

Two chords intersect inside a circle. One chord is divided into segments of 3 cm and 8 cm. The other chord has one segment of 4 cm. The other segment is:

1. 5 cm
2. 6 cm
3. 7 cm
4. 8 cm

Answer: B

Solution: 3 × 8 = 4 × x ⇒ x = 6.

6. Tangent Length

From an external point, two tangents are drawn to a circle. If one tangent is 12 cm long, the other tangent is:

1. 10 cm
2. 11 cm
3. 12 cm
4. Cannot be determined

Answer: C

7. Common Tangents

How many common tangents do two non-intersecting, externally separate circles have?

1. 2
2. 3
3. 4
4. 5

Answer: C

8. Circumference

The circumference of a circle is 44 cm. Its radius is (Take π = 22/7):

1. 6 cm
2. 7 cm
3. 8 cm
4. 9 cm

Answer: B

9. Area Ratio

The radii of two circles are in the ratio 2:5. Their areas are in the ratio:

1. 2:5
2. 4:25
3. 5:2
4. 8:125

Answer: B

10. Sector Area

A circle has radius 14 cm. Find the area of a 90° sector.

1. 49π cm²
2. 98π cm²
3. 196π cm²
4. 154π cm²

Answer: A

11. Angle in a Semicircle

An angle subtended by the diameter at any point on the circle is:

1. 45°
2. 60°
3. 90°
4. 120°

Answer: C

12. Concentric Circles

Two concentric circles have radii 5 cm and 9 cm. The area of the ring between them is:

1. 56π cm²
2. 81π cm²
3. 25π cm²
4. 36π cm²

Answer: A

Solution: π(9² − 5²) = π(81 − 25) = 56π.

13. Arc Length

The radius of a circle is 21 cm. Find the length of a 60° arc.

1. 7π cm
2. 14π cm
3. 21π cm
4. 42π cm

Answer: A

14. Chord Property

The perpendicular drawn from the center of a circle to a chord:

1. Bisects the chord
2. Is parallel to the chord
3. Touches the circumference only
4. Need not bisect the chord

Answer: A

15. Cyclic Quadrilateral

Opposite angles of a cyclic quadrilateral are:

1. Equal
2. Complementary
3. Supplementary
4. Twice each other

Answer: C

Circle Formulas for CAT 2026

Computational formulas only. The theorems come next.

Quantity	Formula
Circumference	2πr
Area	πr²
Arc length (central angle θ°)	(θ/360) × 2πr
Sector area	(θ/360) × πr²
Segment area	Sector area − triangle area
Chord at distance d from centre	2√(r² − d²)
Chord subtending angle θ at centre	2r sin(θ/2)
Tangent length from external point	√(d² − r²)

Circles in coordinate geometry. Expect roughly one coordinate geometry question a year, and circles are its most common subject.

Form	Expression
Centre–radius form	(x − h)² + (y − k)² = r²
General form	x² + y² + 2gx + 2fy + c = 0
Centre from general form	(−g, −f)
Radius from general form	√(g² + f² − c)
Line is a tangent when	distance from centre to line = r
Two circles touch externally when	distance between centres = r₁ + r₂
Two circles touch internally when	distance between centres = |r₁ − r₂|

Circle Theorems for CAT 2026: The 15 Properties That Solve Everything

This set is closed. Learn these CAT Circles Questions and you have the machinery for essentially every circles question CAT has asked.

Tangents

1. A radius is perpendicular to the tangent at the point of contact.
2. Two tangents from an external point are equal in length.
3. Tangent length from an external point = √(d² − r²).
4. Alternate segment theorem - the angle between a tangent and a chord equals the inscribed angle in the alternate segment.

Chords

1. Equal chords subtend equal angles at the centre and are equidistant from it.
2. A perpendicular from the centre to a chord bisects it, and the converse holds.
3. Intersecting chords theorem - for chords meeting at P inside a circle, AP × PB = CP × PD.
4. Power of a point - for a secant and tangent from an external point, PT² = PA × PB.

Angles

1. Inscribed angle theorem - the angle at the centre is twice the angle at the circumference subtending the same arc.
2. An angle in a semicircle is 90°. The most frequently used property in CAT circles PYQs.
3. Angles in the same segment are equal.
4. Two secants meeting outside the circle: the angle equals half the difference of the intercepted arcs.
5. Two secants meeting inside the circle: the angle equals half the sum of the intercepted arcs.

Cyclic quadrilaterals

1. Opposite angles sum to 180°, and an exterior angle equals the interior opposite angle.
2. Ptolemy's theorem - for cyclic quadrilateral ABCD, AC × BD = (AB × CD) + (BC × AD).

Concyclic test: if a segment subtends equal angles at two points on the same side of it, all four points lie on one circle. Most useful in reverse - spotting a hidden circle in a question that never mentions one.

Types of CAT Circles Questions 2026

Type	The trap
Tangent	Forgetting the radius–tangent right angle, which is usually the only route in
Chord	Confusing a chord with a secant. Intersecting chords applies inside; power of a point applies outside
Angle in circle	Mixing the angle at the centre with the angle at the circumference. One is twice the other, and both appear as options
Sector	Forgetting that segment area needs the triangle subtracted from the sector
Arc	Reading arc length as chord length
Cyclic quadrilateral	Not noticing the quadrilateral is cyclic. Nothing in the question will say so
Coordinate geometry	Sign error reading the centre. In x² + y² + 2gx + 2fy + c = 0 the centre is (−g, −f), not (g, f)

How to Solve CAT Circles Questions 2026 Step by Step

Step 1 - Draw it, even when a figure is given. Redraw at a size you can annotate. Most circles errors trace to a cramped diagram, not a missing formula.

Step 2 - Mark everything equal. Equal radii, equal tangents from a point, equal chords. Do this before rereading the question. Half the time the answer becomes visible.

Step 3 - Name the theorem out loud. Not "use circle properties" but "two chords meeting inside, so intersecting chords." If you cannot name it in fifteen seconds, hunt for a hidden right angle or a hidden cyclic quadrilateral. One of those is usually the way in.

Step 4 - Add the construction. Draw the radius to the point of tangency. Join the centre to the chord's midpoint. Circles questions are often solved by a line that was not in the original figure - usually one that creates a right triangle.

Step 5 - Simplify, then sanity-check. A radius longer than a diameter, or an angle above 180°, means you have swapped a centre angle for a circumference angle CAT circles questions PDF.

The habit underneath all five: circles questions convert into triangle questions. Nearly every solution runs through a right triangle you created. When stuck, the question is rarely "which circle formula" - it is "where is the right triangle".

Solved CAT Circles Questions 2026 With Answers and Explanations

Level 1: Single-Theorem Questions

Q1. From an external point P, the distance to the centre of a circle of radius 5 cm is 13 cm. Find the length of the tangent from P.

Q2. AB is a diameter of a circle and C lies on the circle. If ∠BAC = 35°, find ∠ABC.

Q3. A chord lies 8 cm from the centre of a circle of radius 17 cm. Find its length.

Solutions

Q1. 12 cm. √(13² − 5²) = √144 = 12. One formula, no working.

Q2. 55°. The angle in a semicircle is 90°, so ∠ACB = 90°. Then ∠ABC = 180 − 90 − 35 = 55°. The moment a question names a diameter and a point on the circle, you have a right triangle for free.

Q3. 30 cm. Half-chord = √(17² − 8²) = √225 = 15, so the chord is 30 cm. The perpendicular from the centre bisects the chord — that is the whole question.

Level 2: Two-Concept Questions

Q4. Chords AB and CD meet at P inside a circle. AP = 6 cm, PB = 4 cm, CP = 3 cm. Find PD.

Q5. ABCD is a cyclic quadrilateral with ∠ADC = 105°. Side AB is extended to E. Find ∠CBE.

Q6. Tangents PA and PB are drawn from an external point P to a circle with centre O. If ∠APB = 50°, find ∠AOB.

Solutions

Q4. 8 cm. AP × PB = CP × PD gives 24 = 3 × PD.

Q5. 105°. An exterior angle of a cyclic quadrilateral equals the interior opposite angle. No calculation at all — but only if you spot that the quadrilateral is cyclic, which the question does not tell you.

Q6. 130°. OAPB is a quadrilateral with right angles at A and B, since a radius meets a tangent perpendicularly. So ∠AOB = 360 − 90 − 90 − 50 = 130°.

Level 3: Circles Combined With Triangles and Quadrilaterals

Q7. Two circles of radii 9 cm and 4 cm touch externally. Find the length of their direct common tangent.

Q8. A circle is inscribed in a right triangle with legs 6 cm and 8 cm. Find its radius.

Q9. For the circle x² + y² − 6x + 4y − 12 = 0, find the centre and radius, then determine whether 3x − 4y + 7 = 0 is a tangent to it.

Solutions

Q7. 12 cm. The centres are r₁ + r₂ = 13 cm apart. Direct common tangent = √(d² − (r₁ − r₂)²) = √(169 − 25) = 12.

Q8. 2 cm. The hypotenuse is 10. For a right triangle, r = (a + b − c)/2 = (6 + 8 − 10)/2 = 2. Check it with area = r × s: 24 = r × 12.

Q9. Centre (3, −2), radius 5, and the line is not a tangent. From 2g = −6 and 2f = 4 we get g = −3, f = 2, so the centre is (−g, −f) = (3, −2) and the radius is √(9 + 4 + 12) = 5. Distance from the centre to the line = |9 + 8 + 7| / 5 = 4.8. Since 4.8 < 5, the line cuts the circle - it is a secant. Most students stop at computing the distance and assume tangency; the comparison is the question.

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Previous Year CAT Circles Questions

The Four Configurations CAT Circles Questions Keeps Reusing

Circles PYQs recycle a small set. Solving the CAT mocks will also help in letting us know what type of questions are asked.

1. Two circles touching externally with a common tangent. Join the centres, drop a perpendicular, solve the right triangle. Sometimes extended to a third circle.
2. A circle inscribed in a triangle, square or rhombus. Set up in terms of r, then use area = r × s.
3. Intersecting chords with a ratio condition. The theorem plus one algebraic relation, almost always a quadratic.
4. A diameter subtending 90°. The most reused property in CAT circles history.

If your hours on this topic are limited, spend them here rather than on breadth.

P1. Two circles each of radius 4 cm touch externally. A third circle touches both externally, and all three share a common tangent. Find the radius of the third circle.

P2. AB is a diameter of a circle of radius 5 cm. P lies on the circle with PB = 6 cm. Find AP.

P3. Chords AB and CD meet at E inside a circle. AB = 12 cm, CE = 4 cm and ED = 8 cm. Find |AE − EB|.

Solutions

P1. 1 cm. Put the tangent on the x-axis. The two large centres sit at height 4, the small one at height x. Horizontal separation between a large centre and the small one is found from the right triangle: 4² + (4 − x)² = (4 + x)², which gives 16 = 16x, so x = 1. Configuration 1, with the extra circle.

P2. 8 cm. AB = 10 and ∠APB = 90°, so AP = √(100 − 36) = 8. Configuration 4.

P3. 4 cm. Let AE = x, so EB = 12 − x. Then x(12 − x) = 32, giving x² − 12x + 32 = 0 and x = 4 or 8. The two segments are 4 and 8 either way, so the difference is 4. Configuration 3 - note the quadratic yields both segments at once, which is why the answer is unambiguous.

Tricks to Solve Circle Questions 2026 Faster

Draw extra radii. The most productive single move in circles. A radius to a point of tangency creates a right angle; a radius to a chord's endpoint creates an isosceles triangle. Both give you something to work with.

Convert to a triangle. Almost every circles solution runs through a triangle. If you are stuck, find the triangle rather than another formula.

Use symmetry. Two circles touching, a circle inscribed in a square, three equal circles - the configuration is symmetric, so work with half of it and double at the end.

Look for the hidden cyclic quadrilateral. If two opposite angles of a quadrilateral sum to 180°, it is cyclic and the whole theorem set becomes available. Questions rarely announce this.

Sanity-check against the diameter. Any length you compute inside a circle must be at most the diameter. A surprising number of errors are caught in five seconds this way.

Common Mistakes to Avoid while Solving CAT Circles Questions 2026

• Forgetting the radius–tangent right angle. Symptom: you cannot find a way into a tangent question. That right angle is usually the only entry point.
• Confusing a chord with a secant. Symptom: applying AP × PB = CP × PD to an external point. Inside the circle it is intersecting chords; outside it is power of a point.
• Missing that a quadrilateral is cyclic. Symptom: a question looks under-specified. Check whether all four vertices lie on the circle.
• Swapping centre and circumference angles. Symptom: your answer is exactly double or exactly half an option. That is the signature of this error.
• A cramped diagram. Symptom: you keep re-reading the question. Redraw it larger.

Conclusion

Mastering CAT Circles Questions is not about memorizing dozens of formulas - it's about understanding the core concepts and applying them with confidence. From tangents and chords to cyclic quadrilaterals and angle properties, consistent practice will help you solve even the trickiest Geometry problems with ease. Make it a habit to practise Circles Questions for CAT through previous year papers, sectional tests, and timed mocks to improve both speed and accuracy.

Finally, keep revising with the CAT circles questions PDF before the exam to reinforce key formulas, shortcut techniques, and exam-level questions. With the right preparation, this topic can become one of your strongest scoring areas in CAT 2026.