Area and Perimeter

The regular Class IX course links perimeter with the geometry of circles, then develops area through sectors, segments, Heron’s formula and its generalisation for cyclic quadrilaterals.

Core ideas

Perimeter measures boundary length. For a circle, circumference C=2πr=πd, so π is the ratio C/d. It is irrational: its decimal expansion does not terminate or repeat exactly. For an arc subtending θ degrees at the centre, arc length = (θ/360°)×2πr.

Area measures the region enclosed by a shape. A sector is a fraction θ/360° of the circle, so its area is (θ/360°)πr². The area of a minor segment is the sector area minus the area of the triangle formed by the two radii and chord.

Heron’s formula: for triangle sides a,b,c and semiperimeter s=(a+b+c)/2, area = √[s(s−a)(s−b)(s−c)]. Brahmagupta’s formula: for a cyclic quadrilateral with sides a,b,c,d and semiperimeter s, area = √[(s−a)(s−b)(s−c)(s−d)]. If one side tends to zero, the cyclic quadrilateral formula reduces to Heron’s triangle formula.

Worked examples

Arc length

A 60° arc of a circle of radius 21 cm has length (60/360)×2×(22/7)×21 = 22 cm.

Heron’s formula

A triangle has sides 13 cm, 14 cm and 15 cm. Its semiperimeter is 21 cm. Area = √[21×8×7×6] = √7056 = 84 cm².

Sector area

A 90° sector of radius 8 cm has area (90/360)π×64 = 16π cm².

Practice questions with answers

Find the circumference of a circle with diameter 14 cm. Use π=22/7.

C=πd=(22/7)×14=44 cm.

A 120° sector has radius 6 cm. Find its area.

(120/360)×π×36=12π cm².

Why is Heron’s formula a special case of Brahmagupta’s formula?

Set one side of the cyclic quadrilateral to 0. Then its semiperimeter is the triangle’s s and the expression becomes √[s(s−a)(s−b)(s−c)], Heron’s formula.

Practical investigation

Measure a circular lid’s diameter and circumference at least three times with a flexible tape or thread. Calculate C/d each time and compare the estimates. Record measurement error and explain why the results cluster near π.

For a sector, draw a circle, mark a central angle and compare its fraction of 360° with the fraction of the full circumference and area.

These are original revision notes aligned to the CBSE Class IX 2026–27 curriculum. Use them with your prescribed NCERT book and your teacher’s directions. Official CBSE syllabus · Official NCERT textbook resource