Measuring Space: Perimeter and Area Class 9 Notes

Measuring Space: Perimeter and Area is the Class 9 chapter on how far it is around a shape and how much space it covers. This guide goes through perimeter, the story of π, arc length, areas of triangles and parallelograms, Heron's and Brahmagupta's formulas, Baudhāyana's squaring of a rectangle, and circles and sectors. Every formula comes with a fully worked example and the answers are checked step by step.

Perimeter and the idea of π

Perimeter is the total length around the border of a shape: the distance a tiny insect walks to go once around and return to its start.

A square is a rectangle with a = b, so 2(a + a) = 4a. For every square, perimeter : side = 4 : 1; for every equilateral triangle it is 3 : 1. Double the side of an equilateral triangle and the perimeter doubles too: 3(2a) = 6a.

The perimeter of a circle is its circumference (C). For every circle, C ÷ D (D is the diameter) is the same number, called π (pi). So C = πD = 2πr.

The history of π

People have estimated π with better and better accuracy:

π is irrational: it cannot be written as a fraction of two integers, and its decimal goes on forever without repeating. We use 22/7 or 3.14 because they are close enough for most calculations, not because they are exact.

Length of an arc

An arc is part of the circumference. For an arc that makes angle θ at the centre:

Arc length = 2πr × (θ/360)

• Semicircle (180°): πr
• Quarter circle (90°): πr/2

Worked example: r = 7 cm, θ = 60°. Arc = 2 × (22/7) × 7 × (60/360) = 44 × 1/6 = 44/6 ≈ 7.33 cm.

Perimeter of a semicircle = curved part + diameter = πr + 2r. For r = 7 cm: 22 + 14 = 36 cm.

Athletics tracks use staggered starts because outer lanes curve along arcs of bigger radius. Moving the outer starting points forward makes every runner cover exactly 400 m.

Area of parallelograms and triangles

Area is the space a shape covers, measured in square units.

• Parallelogram: cut and rearrange it into a rectangle with the same base b and height h. Area = b × h.
• Triangle: two congruent copies join to make a parallelogram of base b and height h, so one triangle = (1/2) × b × h.

So a triangle and a parallelogram with the same base and height have areas in the ratio 1 : 2.

Median theorem: a median divides a triangle into two triangles of equal area. They may not be congruent, but they have equal bases and the same height.

Heron's formula

When you know only the three sides a, b and c:

1. Semi-perimeter s = (a + b + c)/2
2. Area = √s(s − a)(s − b)(s − c)

Worked example (3, 4, 5): s = 12/2 = 6. Area = √6 × 3 × 2 × 1 = √36 = 6 sq. units. Check with (1/2) × 3 × 4 = 6.

Worked example (13, 14, 15): s = 42/2 = 21. Area = √21 × 8 × 7 × 6 = √7056 = 84 sq. units.

Heron's formula also works for an equilateral triangle: with s = 3a/2 it gives (√3/4)a².

Two more area formulas use the circumradius R and inradius r: Area = abc/(4R) and Area = r(a + b + c)/2 = rs.

Brahmagupta's formula

You cannot find the area of a general quadrilateral from its sides alone, but you can for a cyclic quadrilateral (all four vertices on one circle). In 628 CE, Brahmagupta gave, for sides a, b, c, d and s = (a + b + c + d)/2:

Area = √(s − a)(s − b)(s − c)(s − d)

• Link to Heron: put d = 0 and it becomes √s(s − a)(s − b)(s − c).
• Check with a rectangle (always cyclic) of sides 3 and 4: s = (3 + 4 + 3 + 4)/2 = 7. Area = √4 × 3 × 4 × 3 = √144 = 12. And 3 × 4 = 12.

Baudhāyana's squaring of a rectangle

To square a shape means to build a square with the same area. In the Śhulbasūtra, Baudhāyana gave a construction for a rectangle with sides a and b (a > b). It is a picture of the identity:

ab = (a + b)/2² − (a − b)/2²

The construction draws a square of side (a + b)/2, then uses the Baudhāyana–Pythagoras theorem to cut away a square of side (a − b)/2. What remains equals ab, so a square on the final side has the same area as the rectangle.

Check: a = 8, b = 2. (10)/2² − (6)/2² = 25 − 9 = 16 = 8 × 2. A square of side 4 has the same area.

Area of a circle and a sector

Archimedes showed a circle's area equals a right triangle with legs r and 2πr: (1/2) × r × 2πr = πr². Nīlakaṇṭha Somayājī's picture proof slices the circle into thin wedges; rearranged, they approach a parallelogram of base πr and height r, giving πr².

Worked example: r = 14 cm. Area = (22/7) × 14 × 14 = 22 × 2 × 14 = 616 sq. cm.

Doubling the radius makes the area four times: π(2r)² = 4πr².

A sector is a pie-slice bounded by two radii and an arc: Area = πr² × (θ/360). A quadrant is πr²/4. A segment (between a chord and its arc) = sector area − triangle area.

Worked example: r = 6 cm, θ = 60°. Area = 36π × 1/6 = 6π ≈ 6 × 3.14 = 18.84 sq. cm.

Remember this

• Circumference = 2πr; arc = 2πr × θ/360; sector = πr² × θ/360.
• Triangle = (1/2)bh; parallelogram = bh.
• Heron: find s first, then √s(s − a)(s − b)(s − c).
• Brahmagupta works only for cyclic quadrilaterals.
• 22/7 and 3.14 are approximations of π.

Important questions with answers

1. Find the arc length for r = 14 cm, θ = 90°. 2 × (22/7) × 14 × 1/4 = 88 × 1/4 = 22 cm.

2. Find the area of a circle of radius 7 cm. (22/7) × 7 × 7 = 154 sq. cm.

3. Find the area of a triangle with sides 5, 12 and 13 cm. s = 15. Area = √15 × 10 × 3 × 2 = √900 = 30 sq. cm.

4. Find the area of an equilateral triangle of side 6 cm using Heron's formula. s = 9. Area = √9 × 3 × 3 × 3 = √243 = 9√3 sq. cm. This matches (√3/4) × 36 = 9√3.

5. Why does a median split a triangle into equal areas? The two parts have equal bases (the median meets the midpoint) and the same height.

6. Find the area of a sector of radius 14 cm and angle 90°. 616 × 1/4 = 154 sq. cm.

7. What happens to Brahmagupta's formula when d = 0? It becomes Heron's formula, √s(s − a)(s − b)(s − c).

Common mistakes to avoid

• Using the full perimeter instead of the semi-perimeter s in Heron's formula.
• Forgetting the diameter when finding the perimeter of a semicircle.
• Using Brahmagupta's formula for a quadrilateral that is not cyclic.
• Writing area in cm instead of sq. cm.

For guided practice on these formulas, study this chapter with Joy on Learnijoy.