India's #1 AI Tutorformula-sheet · Mathematics · Chapter 5हिंदी में पढ़ें →

Class 9 Mathematics Chapter 5 I'm Up and Down, and Round and Round — Formulas & Key Points

Chapter 5 of NCERT Class 9 Mathematics introduces circles through their defining property: all points on a circle are equidistant from a fixed centre. This chapter builds theorems about chords, angles subtended by arcs, cyclic quadrilaterals, and the relationship between central and inscribed angles. Mastering the formulas and properties here is crucial for both board exams and higher geometry. Below is a structured formula sheet with tables, worked examples, and quick-revision aids.

Your child's private AI tutor — trained on NCERT.

3-day free trial · ₹1 to start · Cancel anytime.

Start 3-day free trial →

Key takeaways

✓A circle is the locus of all points equidistant from a fixed centre; radius defines this distance.
✓Perpendicular from centre to a chord bisects the chord; equal chords are equidistant from centre.
✓Angle subtended by an arc at the centre is exactly twice the angle at any point on the circle.
✓In a cyclic quadrilateral, opposite angles always sum to 180 degrees.
✓Any angle inscribed in a semicircle is a right angle (90 degrees).
✓Equal chords subtend equal angles at the centre; conversely, equal central angles mean equal chords.
✓Longer chords lie closer to the centre; use d = √(r² − l²/4) to relate chord length, radius, and distance.

Core Definitions and Terminology

Before diving into formulas, anchor yourself in the precise language of circles. A circle is not just a round shape — it is the set of all points in a plane at a fixed distance (radius) from a fixed point (centre). A chord is any line segment joining two points on the circle. The diameter is a special chord passing through the centre, and it is always the longest chord. An arc is a continuous piece of the circle between two points; every chord divides the circle into a minor arc (shorter) and a major arc (longer). The perpendicular distance from the centre to a chord is the shortest distance, measured along the line perpendicular to that chord. Understanding these definitions ensures you apply formulas correctly and interpret geometry problems without confusion.

Circle: locus of points at distance r from centre O
Radius: line segment from centre to any point on circle; all radii equal
Diameter: chord through centre; d = 2r; longest possible chord
Chord: line segment joining any two points on the circle
Arc: curved portion of circle between two points
Perpendicular distance: shortest distance from centre to chord, measured perpendicularly

Essential Formulas and Properties — Master Table

Every theorem in this chapter translates into a testable formula or property. The table below consolidates all key results. Memorise the 'When to Use' column — it tells you which property to invoke when you see equal chords, equal angles, or perpendiculars. For instance, if a question states two chords are equal, immediately recall they are equidistant from the centre and subtend equal angles at the centre. If you see a semicircle, remember the inscribed angle is 90 degrees. These logical links save time and prevent errors under exam pressure. Refer to this table during practice and before exams.

Formula Table: Chords and Centre

The relationship between chords and the centre is governed by perpendicularity and distance. When the perpendicular from centre meets a chord, it bisects that chord into two equal halves. Conversely, if a line from the centre bisects a chord, it must be perpendicular to that chord. Equal chords are equidistant from the centre, meaning the perpendicular distances are identical. If you know radius r, chord length l, and perpendicular distance d, they satisfy the Baudhāyana–Pythagoras relation: (l/2)² + d² = r². Rearranging gives l = 2√(r² − d²) or d = √(r² − l²/4). Longer chords correspond to smaller d (closer to centre); shorter chords have larger d (farther from centre). Use these formulas whenever a problem gives you two of the three quantities and asks for the third.

Perpendicular from centre to chord bisects the chord
Equal chords ⇔ equidistant from centre
Chord length: l = 2√(r² − d²)
Distance from centre: d = √(r² − (l/2)²)
Longer chord ⇒ smaller d; shorter chord ⇒ larger d

Formula Table: Angles at Centre and on Circle

The angle subtended by an arc at the centre of the circle is exactly double the angle subtended by the same arc at any point on the remaining part of the circle. Mathematically, if arc PQ subtends angle θ at centre O and angle α at point R on the circle, then θ = 2α. This is the Inscribed Angle Theorem. A critical corollary: angles subtended by the same arc at different points on the circle are all equal (each is half the central angle). Another special case: a semicircular arc (diameter) subtends 180° at centre, so the inscribed angle is 90°. Hence any angle in a semicircle is a right angle. Use this property whenever you see a diameter or need to prove a right angle in circle geometry.

Central angle = 2 × inscribed angle (for same arc)
Angles in the same segment (same arc, different points on circle) are equal
Angle in a semicircle = 90° (diameter subtends straight angle at centre)
If ∠AOB = θ, then ∠APB = θ/2 (P on major arc)

Formula Table: Equal Chords and Equal Angles

Two chords of the same circle are equal in length if and only if they subtend equal angles at the centre. This bi-conditional statement is powerful: proving chords equal is equivalent to proving their central angles equal. Why? Because the triangles formed by joining the centre to the chord endpoints are isosceles (radii are equal). If two such triangles have all three sides equal (SSS congruence), their apex angles (at centre) are equal. Conversely, if apex angles are equal and two sides (radii) are equal, the triangles are congruent (SAS), so the third sides (chords) are equal. Use this property to convert angle conditions into length conditions and vice versa in proofs.

Equal chords ⇔ equal angles at centre
If AB = CD (chords), then ∠AOB = ∠COD
If ∠AOB = ∠COD, then chord AB = chord CD
Based on congruence of isosceles triangles (radii equal)

Cyclic Quadrilaterals: Opposite Angles Sum to 180°

A quadrilateral is cyclic if all four vertices lie on the same circle. The defining property: the sum of each pair of opposite angles is 180 degrees. Formally, in cyclic quadrilateral ABCD, ∠A + ∠C = 180° and ∠B + ∠D = 180°. Why? Each angle is an inscribed angle subtending the arc opposite to it. The two opposite arcs together make a full circle (360°), so the central angles sum to 360°, and the inscribed angles sum to 180°. This property is both a test for cyclicity and a tool for finding unknown angles. If opposite angles do not sum to 180°, the quadrilateral cannot be inscribed in a circle. Conversely, if you know three angles, the fourth is determined.

Cyclic quadrilateral: all vertices on one circle
Opposite angles sum to 180°: ∠A + ∠C = 180°, ∠B + ∠D = 180°
Test for cyclicity: check if opposite angle pairs sum to 180°
Given three angles, fourth angle = 180° − opposite angle

Memory Tricks and Mnemonics

Formulas stick better with mental hooks. For perpendicular-bisector property, visualise folding a paper circle along a chord — the crease (perpendicular from centre) splits the chord exactly in half. For the inscribed-angle theorem, remember 'centre sees double': the angle at the centre is always twice what the circle-point sees. For cyclic quadrilaterals, think 'opposite angles are supplementary' (sum to 180°) — if you push one angle up, the opposite must drop to keep the sum constant. For the semicircle right-angle, recall Thales' theorem: any triangle inscribed in a semicircle with the diameter as base is right-angled at the third vertex. Use these images during problem-solving to trigger the correct property instantly.

'Perpendicular from centre bisects chord' — fold-and-split memory
'Centre sees double' — central angle = 2 × inscribed angle
'Opposite angles add to 180' — cyclic quadrilateral mnemonic
'Semicircle = right angle' — Thales' theorem visual
'Equal chords, equal angles at centre' — symmetry principle

Common Mistakes and How to Avoid Them

Students often confuse central angle with inscribed angle — always check which vertex the angle is measured from. Another frequent error: assuming any quadrilateral in a circle is cyclic; verify all four vertices lie on the circle, not just three. When using the chord-distance formula l = 2√(r² − d²), remember d is perpendicular distance, not any distance. Sign errors creep in when rearranging; always square both sides carefully. Also, do not assume two chords are equal just because they look equal in a diagram — prove it using angle or distance properties. Finally, in cyclic quadrilaterals, students sometimes add adjacent angles instead of opposite; label vertices clearly and double-check which angles are opposite before summing to 180 degrees.

Central vs inscribed angle: check the vertex (centre or circle point)
Cyclic quadrilateral: verify all four points on circle, not just three
Perpendicular distance d is shortest distance, measured at right angle to chord
Square carefully in chord formula to avoid sign and arithmetic errors
Opposite angles sum to 180° in cyclic quad, not adjacent angles

Solved Mini-Examples Applying the Formulas

Concrete examples anchor abstract formulas in your mind and show you the step-by-step method for exam-style questions. Work through these three problems carefully, noting which property or formula is invoked at each step. Example 1 uses the chord-distance formula. Example 2 applies the inscribed-angle theorem. Example 3 tests cyclic-quadrilateral opposite-angle property. After studying these, try similar problems from NCERT exercise 5.1 to 5.3 to build fluency and speed. These worked solutions also model the layout and reasoning you should present in board exams for full marks — clearly state the given, the formula used, the substitution, and the final answer with units where applicable. Practice writing solutions this way to maximise your score even if you make a small arithmetic slip.

One-Glance Last-Minute Revision Box

In the final minutes before your exam, scan this box to refresh all critical formulas and theorems. Circle definition: all points at distance r from centre. Perpendicular from centre bisects chord. Equal chords are equidistant from centre and subtend equal central angles. Central angle equals twice the inscribed angle for the same arc. Angle in semicircle is always 90 degrees. Cyclic quadrilateral: opposite angles sum to 180 degrees. Chord length formula: l equals two times the square root of r squared minus d squared. Use the Baudhāyana–Pythagoras theorem in the right triangle formed by radius, half-chord, and perpendicular distance. Keep this box bookmarked on your phone or printed in your notebook for quick revision during study breaks, on the way to the exam centre, or right before entering the hall. Pair it with CBSETUTOR.ai's 24×7 AI tutor for instant doubt-clearing at ₹999/month (all classes 6-12, 3-day free trial) — upload a photo of any problem and get step-by-step solutions on your phone anytime, anywhere.

Circle: r = radius, all points equidistant from O
Perpendicular bisects chord; equal chords ⇔ equidistant from centre
Central ∠ = 2 × inscribed ∠; angle in semicircle = 90°
Cyclic quad: ∠A + ∠C = 180°, ∠B + ∠D = 180°
Chord formula: l = 2√(r² − d²), d = √(r² − l²/4)
Equal chords ⇔ equal central angles; congruent isosceles triangles

Frequently asked questions

What is the definition of a circle in CBSE Class 9 Mathematics Chapter 5?+

A circle is the set of all points in a plane that are at a fixed distance (radius) from a fixed point (centre). This equidistant property is the mathematical essence of a circle.

How do I prove that the perpendicular from the centre bisects a chord?+

Draw radii to both endpoints of the chord, forming an isosceles triangle (two radii are equal). The perpendicular from the apex (centre) to the base (chord) always bisects the base in an isosceles triangle.

What is the formula relating chord length, radius, and perpendicular distance?+

Use l = 2√(r² − d²), where l is chord length, r is radius, and d is perpendicular distance from centre to chord. This comes from the Baudhāyana–Pythagoras theorem in the right triangle formed.

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

A semicircle corresponds to a diameter, which subtends 180° at the centre. By the inscribed-angle theorem, any point on the circle sees the diameter at half that angle: 180°/2 = 90°.

How do I test if a quadrilateral is cyclic?+

Check if the sum of each pair of opposite angles equals 180°. If ∠A + ∠C = 180° and ∠B + ∠D = 180°, the quadrilateral is cyclic; otherwise it cannot be inscribed in a circle.

What is the inscribed-angle theorem in simple terms?+

An arc subtends an angle at the centre that is exactly double the angle it subtends at any point on the remaining circle. Central angle = 2 × inscribed angle for the same arc.

Do equal chords always subtend equal angles at the centre?+

Yes. Equal chords correspond to congruent isosceles triangles (radii are equal), so their apex angles at the centre are equal. Conversely, equal central angles imply equal chords.

Are longer chords closer to or farther from the centre?+

Longer chords are closer to the centre. The formula d = √(r² − l²/4) shows that as chord length l increases, perpendicular distance d decreases, bringing the chord nearer the centre.

Can CBSETUTOR.ai help me solve circle geometry problems instantly?+

Yes. CBSETUTOR.ai offers a 24×7 AI tutor at ₹999/month (one price for classes 6-12). Upload a photo of any circle problem and get step-by-step solutions on your phone. 3-day free trial available.

How do I find the fourth angle in a cyclic quadrilateral if three angles are given?+

Use the property that opposite angles sum to 180°. If you know ∠A, ∠B, ∠C, then ∠D = 180° − ∠B (or equivalently, check ∠A + ∠C should equal 180°, then ∠D follows).

Related resources

CBSE Class 9 Mathematics Chapter 5 Introduction to Euclid's Geometry Worksheet with Answers Class 9 Mathematics Chapter 5 Introduction to Euclid's Geometry — Formulas & Key Points Important Questions: CBSE Class 9 Mathematics Chapter 5 Introduction to Euclid's Geometry CBSE Class 9 Mathematics Chapter 5 I'm Up and Down, and Round and Round Worksheet with Answers CBSE Class 9 Mathematics Chapter 1 Number Systems — Notes CBSE Class 9 Mathematics — Number Systems: complete chapter guide NCERT Solutions for Class 9 Mathematics Chapter 1: Number Systems – Complete Guide Class 9 Mathematics Chapter 2 Polynomials — Formulas & Key Points

Keep learning — related guides

Class 9Mathematics

CBSE Class 9 Mathematics Chapter 8 Predicting What Comes Next: Exploring Sequences and Progressions: mind map & revision

Class 9Mathematics

CBSE Class 9 Mathematics — Predicting What Comes Next: Exploring Sequences and Progressions: complete chapter guide

Class 9Mathematics

CBSE Class 9 Mathematics Chapter 8 Predicting What Comes Next: Exploring Sequences and Progressions — Notes

Class 9Mathematics

NCERT Solutions for CBSE Class 9 Mathematics Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions

Class 9Mathematics

CBSE Class 9 Mathematics — The Mathematics of Maybe: Introduction to Probability: complete chapter guide

Class 9Mathematics

CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability — Notes

Ready to give your Class 9 child the tutor that never sleeps?

CBSETUTOR.ai covers every chapter in the Class 9 NCERT syllabus — Maths, Science, Social Science, English, Hindi and more. 24×7. Patient. Unlimited. 3-day free trial.

Start your child's 3-day free trial →