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Class 10 Mathematics Chapter 10 Circles — Formulas & Key Points

Chapter 10 Circles is one of the highest-scoring chapters in CBSE Class 10 Mathematics, typically carrying 6 to 8 marks in board exams. This formula sheet organizes every theorem, property, definition and worked example you need for quick revision. The chapter explores the elegant relationship between chords, angles and arcs, introducing the angle-subtended theorems and cyclic quadrilaterals. Use this page to memorize formulas, avoid common sign errors, and master the three core theorem applications before your exam.

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Key takeaways

✓The angle subtended by a chord at the centre is exactly twice the angle subtended at any point on the circle (same arc): ∠AOB = 2∠ACB.
✓An angle in a semicircle (formed by a diameter) is always 90°, a critical property for construction and proof problems.
✓Opposite angles in a cyclic quadrilateral are supplementary (sum to 180°), and the converse is also true.
✓Angles in the same segment of a circle are equal, making it easier to solve for unknown angles when multiple points lie on the same arc.
✓Cyclic quadrilaterals are quadrilaterals with all four vertices on a single circle; not all quadrilaterals are cyclic.
✓Memory trick: 'Centre Doubles the Edge' — the central angle is double the circumference angle.
✓Always verify your cyclic quadrilateral solutions by checking both pairs of opposite angles sum to 180°.

All Formulas & Theorems at One Place

Below is a consolidated table of every formula, theorem and key relationship from NCERT Class 10 Mathematics Chapter 10 Circles. Each entry includes the mathematical statement, the conditions under which it applies, and a brief note on when to use it in problem-solving. These theorems form the backbone of every geometry proof and numerical in this chapter, and they recur in competitive exams like Olympiads and entrance tests. Make sure you can state each theorem from memory and recognize the diagram setup instantly. The angle-subtended theorems are the most frequently tested, appearing in 60 to 70 percent of board exam questions on Circles.

Core Theorems Table

This table lists the four fundamental theorems and one key property. Memorize the left column (theorem name) and the middle column (mathematical statement) together. The right column tells you the typical problem scenario where you apply each theorem. For instance, whenever you see a diameter mentioned, immediately think of the 90-degree angle in a semicircle. Whenever a quadrilateral has all four vertices on a circle, check if opposite angles sum to 180 degrees. These pattern-recognition skills will save you minutes in the exam hall and reduce calculation errors.

Angle Subtended by a Chord at Centre | ∠AOB = 2 × ∠ACB | Use when given central angle and need circumference angle, or vice versa; O is centre, C is any point on circle (same arc as A, B).
Angles in the Same Segment are Equal | ∠ACB = ∠ADB (both on same arc) | Use when two or more points lie on the same arc with respect to a chord; helps find unknown angles quickly.
Angle in a Semicircle | ∠ACB = 90° when AB is diameter | Use whenever a diameter is given; the angle opposite the diameter is always a right angle; critical for right-triangle proofs.
Opposite Angles in Cyclic Quadrilateral | ∠A + ∠C = 180°, ∠B + ∠D = 180° | Use for any quadrilateral inscribed in a circle; if one pair of opposite angles is supplementary, the quadrilateral is cyclic.
Converse: Supplementary Opposite Angles ⇒ Cyclic | If ∠A + ∠C = 180°, then ABCD is cyclic | Use to prove that a quadrilateral is cyclic without constructing the circle; check angle sums first.

Key Terms & Definitions

Understanding precise definitions is essential for theorem application and proof writing. CBSE examiners award marks for correct terminology. A chord is any line segment joining two points on the circle, but the diameter is the longest chord passing through the centre. An arc is a curved portion of the circumference between two points; it can be major (longer) or minor (shorter). A segment is the region enclosed by a chord and the arc it cuts off. The central angle is the angle subtended at the centre by an arc or chord, while an inscribed angle (or angle at the circumference) is formed by two chords sharing an endpoint on the circle. A cyclic quadrilateral has all four vertices on the circle (concyclic points). Supplementary angles sum to 180 degrees. The circumcircle is the unique circle passing through all vertices of a polygon. Commit these definitions to memory; they appear in both objective and subjective questions.

Chord: Line segment joining any two points on a circle; diameter is a special chord through the centre.
Arc: Part of the circle's circumference between two points; major arc is longer than 180°, minor arc is shorter.
Segment: Region between a chord and the arc it cuts off; not to be confused with sector (which includes radii).
Central Angle: Angle subtended by an arc or chord at the centre of the circle; measured in degrees or radians.
Inscribed Angle: Angle formed by two chords meeting at a point on the circle; also called angle at the circumference.
Cyclic Quadrilateral: A quadrilateral with all four vertices on a single circle; not every quadrilateral is cyclic.
Concyclic Points: Points that all lie on the same circle; used to define cyclic polygons.
Circumcircle: The circle passing through all vertices of a polygon; every triangle has a unique circumcircle.

Memory Tricks & Mnemonics

Mnemonics and visual cues help you recall theorems under exam pressure. Use the phrase 'Centre Doubles the Edge' to remember that the angle at the centre is twice the angle at the circumference. For cyclic quadrilaterals, think 'Opposite angles are friends who complete each other to 180'. Visualize the diameter as a 'right-angle maker': any point on the circle looking at a diameter sees a perfect 90-degree corner. When you see the word 'same segment', imagine multiple observers standing on the same side of a fence (the chord) — they all see the same view (equal angles). To remember the converse, flip the statement: if you check two opposite angles and they sum to 180, you have proven the quadrilateral is cyclic without drawing anything. Write these tricks in the margin of your exam rough work; they trigger the correct theorem instantly.

'Centre Doubles the Edge' — central angle is always 2 × inscribed angle on the same arc.
'Opposite Angles are Friends to 180' — in a cyclic quadrilateral, opposite angles sum to 180°.
'Diameter = Right-Angle Maker' — any angle in a semicircle is 90°; diameter guarantees a right angle.
'Same Segment, Same Angle' — all angles from the same arc on a chord are equal.
'Check Opposites for Cyclic' — if opposite angles sum to 180°, the quadrilateral is cyclic (converse theorem).
Draw a quick circle sketch in rough work; label centre O, mark arcs, and shade segments to avoid mixing major and minor arcs.

Common Sign, Unit & Notation Mistakes

Small notation errors cost marks in CBSE board exams. Always write angle symbols clearly: ∠AOB, not just AOB or angle AOB. When stating the angle-subtended theorem, specify 'on the same arc'; if you omit this, the statement becomes ambiguous because the angle on the opposite arc is supplementary, not equal. Never confuse arc and segment: an arc is the curve, a segment is the enclosed area. Do not write '∠AOB = ∠ACB' — the correct relation is ∠AOB = 2∠ACB. In cyclic quadrilateral problems, clearly label which angles are opposite; writing ∠A + ∠B = 180° (adjacent angles) is wrong and loses you full marks. When asked to prove a quadrilateral is cyclic, you must show that one pair of opposite angles sums to 180° and explicitly invoke the converse theorem. Skipping the converse citation loses the reasoning mark. Units are typically degrees (°); never write radians unless the question specifies. Double-check your answer: opposite angles should sum to 180°, and all four angles should sum to 360° in any quadrilateral.

Always write ∠AOB with the angle symbol; omitting it is considered informal and may lose presentation marks.
State 'on the same arc' when applying the angle-subtended theorem; the opposite arc gives a supplementary angle.
Do not confuse segment (area) with arc (curve); segment = chord + arc, sector = two radii + arc.
Never write ∠AOB = ∠ACB; the correct relation is ∠AOB = 2 × ∠ACB (centre angle is double).
Label opposite angles clearly in cyclic quadrilaterals; ∠A + ∠C = 180°, not ∠A + ∠B.
Cite the converse theorem explicitly when proving a quadrilateral is cyclic from angle sums.
Use degrees (°) unless radians are specified; mixing units loses marks in board exams.

Worked Mini-Example 1: Angle at Centre & Circumference

Problem: In a circle with centre O, chord AB subtends an angle of 35° at point C on the major arc. Find (a) the central angle ∠AOB, and (b) the angle ∠ADB where D lies on the minor arc. Solution: (a) Given ∠ACB = 35°. By the theorem, ∠AOB = 2 × ∠ACB = 2 × 35° = 70°. (b) Points C and D lie on opposite arcs. The angles they subtend are supplementary: ∠ADB + ∠ACB = 180°, so ∠ADB = 180° − 35° = 145°. Check: 70° is the central angle; half of 70° is 35° (C on major arc), and 180° − 35° = 145° (D on minor arc). This example shows the most common application: converting between central and inscribed angles, and handling points on opposite arcs.

Worked Mini-Example 2: Cyclic Quadrilateral with Angle Ratios

Problem: PQRS is a cyclic quadrilateral. The angles are in the ratio ∠P: ∠Q: ∠R: ∠S = 3: 5: 7: 9. Find all four angles. Solution: Let the angles be 3x, 5x, 7x, 9x. Sum of angles in any quadrilateral is 360°, so 3x + 5x + 7x + 9x = 360° ⇒ 24x = 360° ⇒ x = 15°. Therefore ∠P = 45°, ∠Q = 75°, ∠R = 105°, ∠S = 135°. Verify cyclic property: ∠P + ∠R = 45° + 105° = 150° (error!). Wait, this does not equal 180°. Re-check the ratio: 3 + 7 = 10, 5 + 9 = 14; these are not equal, so the quadrilateral cannot be cyclic with this ratio. Correct approach: For a cyclic quadrilateral, opposite angle sums must be equal. Let ∠P = 3k, ∠R = 7k, then 3k + 7k = 180° ⇒ k = 18°. So ∠P = 54°, ∠R = 126°. Similarly, ∠Q = 5m, ∠S = 9m, then 5m + 9m = 180° ⇒ m = 12.857° (not integer). This ratio is invalid for a cyclic quadrilateral. Lesson: Always verify that the sum of opposite angle expressions equals 180° before proceeding.

Worked Mini-Example 3: Proving a Quadrilateral is Cyclic

Problem: ABCD is a quadrilateral with ∠A = 72° and ∠C = 108°. Prove that ABCD is cyclic. Solution: Check if opposite angles are supplementary. ∠A + ∠C = 72° + 108° = 180°. By the converse of the cyclic quadrilateral theorem, if one pair of opposite angles sums to 180°, the quadrilateral is cyclic. Hence ABCD is cyclic (all four vertices lie on a circle). Note: You do not need to find ∠B and ∠D to prove this; one pair is sufficient. However, for completeness, ∠B + ∠D must also equal 180°, and since the sum of all angles is 360°, we have ∠B + ∠D = 360° − 180° = 180°, confirming the property. This type of proof question appears frequently in CBSE board exams, typically worth 2 to 3 marks. Always cite the converse theorem explicitly in your answer to earn the reasoning mark.

Important Constants, Values & Standard Angles

In this chapter, there are no transcendental constants like π or e, but certain angle measures recur frequently and should be recognized instantly. The right angle (90°) appears in every semicircle problem. The straight angle (180°) is the sum of supplementary angles (opposite angles in cyclic quadrilaterals, or angles on a straight line). A full rotation around the centre is 360°, which is also the sum of all interior angles in any quadrilateral. Common angle pairs you will encounter: 30° and 150°, 45° and 135°, 60° and 120°, 72° and 108°. Memorize the complementary (sum to 90°) and supplementary (sum to 180°) pairs for quick mental checks. When the question gives an angle like 35°, expect to compute 2 × 35° = 70° (centre) or 180° − 35° = 145° (opposite arc). These mental calculations save time and reduce arithmetic errors.

Right angle: 90° (angle in a semicircle, or perpendicular chords)
Straight angle: 180° (sum of supplementary angles, or sum of opposite angles in cyclic quadrilateral)
Full angle: 360° (sum of all angles in any quadrilateral, or one complete revolution)
Common supplementary pairs: 30°–150°, 45°–135°, 60°–120°, 72°–108°, 80°–100°
Doubling for central angle: 40° × 2 = 80°, 55° × 2 = 110°, 65° × 2 = 130°
Halving for inscribed angle: 100° ÷ 2 = 50°, 140° ÷ 2 = 70°, 160° ÷ 2 = 80°

One-Glance Last-Minute Revision Box

Use this box for a final 5-minute review before entering the exam hall. Cover the page and try to recall each formula and theorem name. Then uncover and verify. This active recall technique significantly improves retention. Print this section or bookmark it on your phone. Revise it every night for one week before the exam. Studies show that spaced repetition (revising the same material over several days) is more effective than cramming the night before. If you can reproduce this box from memory, you are ready for any question CBSE throws at you in Chapter 10 Circles. Time yourself: aim to write out all five theorems and definitions in under three minutes.

∠AOB = 2 × ∠ACB — Angle at centre is double the angle at circumference (same arc)
Angle in semicircle = 90° — AB is diameter ⇒ ∠ACB = 90°
Angles in same segment are equal — ∠ACB = ∠ADB (both on same arc)
Cyclic quadrilateral: ∠A + ∠C = 180°, ∠B + ∠D = 180° (opposite angles supplementary)
Converse: If ∠A + ∠C = 180°, then ABCD is cyclic
Chord: line joining two points on circle; Diameter: longest chord through centre
Arc: part of circumference; Segment: region between chord and arc
Common mistake: Forgetting 'same arc' condition; opposite arc gives supplementary angle
Proof tip: Always cite the converse theorem when proving a quadrilateral is cyclic from angle sums

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AI tutor explains why ∠A + ∠C = 180° and how to write the proof to earn full marks in board exams.
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Designed by IIT graduates and CBSE educators; every solution follows NCERT terminology and CBSE marking schemes.

Frequently asked questions

What is the most important theorem in Class 10 Circles chapter?+

The angle-subtended theorem (∠AOB = 2 × ∠ACB) is the most important. It appears in 60-70% of board exam questions on Circles, either directly or as part of a multi-step proof. Master this theorem first, including the condition that A, C, B lie on the same arc.

How do I remember which angle is double and which is half?+

Use the mnemonic 'Centre Doubles the Edge'. The angle at the centre (O) is always double the angle at the circumference (edge). So central angle = 2 × inscribed angle, and inscribed angle = central angle ÷ 2. Draw a quick sketch to visualize the centre and the point on the circle.

What if the two points are on opposite arcs of the chord?+

If point C is on one arc and point D is on the opposite arc, then ∠ACB and ∠ADB are supplementary: they sum to 180°. This is because the two arcs together make the full circle (360°), so the inscribed angles (each half the respective arc) sum to 180°. Always check which arc the point lies on before applying the theorem.

How can I prove a quadrilateral is cyclic without drawing the circle?+

Measure or calculate one pair of opposite angles. If they sum to 180°, invoke the converse of the cyclic quadrilateral theorem to conclude the quadrilateral is cyclic. You do not need to construct the circle or find the centre; the angle sum is sufficient proof. Always cite the converse theorem in your written answer to earn the reasoning mark.

Why is the angle in a semicircle always 90 degrees?+

A diameter subtends 180° at the centre (it is a straight line through the centre). By the angle-subtended theorem, the angle at any point on the circle is half of 180°, which is 90°. This property holds for any position of the third vertex on the circle (excluding the endpoints of the diameter).

Are all quadrilaterals cyclic?+

No. Only quadrilaterals whose opposite angles sum to 180° are cyclic. For example, a rectangle is cyclic (all angles are 90°, so 90 + 90 = 180), but a general trapezium or random quadrilateral is not. To test, check if ∠A + ∠C = 180° and ∠B + ∠D = 180°.

What is the difference between arc and segment?+

An arc is the curved portion of the circle's circumference between two points. A segment is the region (area) enclosed by the arc and the chord joining the two points. Think of the arc as the boundary and the segment as the filled-in region. Do not confuse segment with sector, which includes two radii.

How many marks does Circles carry in CBSE Class 10 board exams?+

Circles typically carries 6 to 8 marks in the CBSE Class 10 Mathematics board exam. Questions range from 2-mark theorem-based problems to 3 or 4-mark proof questions. Mastering the five core theorems and practicing 10 to 15 problems is enough to secure full marks in this chapter.

Can I use the same theorem for major and minor arcs?+

The angle-subtended theorem applies to points on the same arc. If you move a point from the major arc to the minor arc (or vice versa), the inscribed angle changes to 180° minus the original angle. Always specify 'same arc' in your answer, and draw a sketch to identify whether points are on the same or opposite arcs.

What should I revise one night before the exam for Circles?+

Revise the one-glance box in this page: the five core theorems, the mnemonic 'Centre Doubles the Edge', the 90° angle in semicircle, and the cyclic quadrilateral property (opposite angles sum to 180°). Practice writing one proof from memory. Do not attempt new difficult problems the night before; focus on recalling formulas and theorem statements quickly and accurately.

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