Perimeter and Area — Measuring Space (RBSE Class 9 · Mathematics)

How long is the boundary, and how much surface does it enclose? These two questions — perimeter and area — measure space itself. The chapter's quiet star is π, the constant hiding in every circle.

RBSE note (2026-27). Class 9 uses the new NCF (Ganita Prakash 9) Mathematics textbook; this chapter is Measuring Space: Perimeter and Area. BSER (Ajmer) sets the exam.

1. Perimeter of a shape

The perimeter is the total length of the boundary of a closed figure — add up all the sides.

• Square (side a): $P=4a$
• Rectangle (l, b): $P=2(l+b)$
• Triangle (a, b, c): $P=a+b+c$
• Regular polygon (n sides, side a): $P=na$

Unit: same as length (cm, m). Area is measured in square units (cm², m²).

2. The circumference of a circle and π

Measure the circumference (C) and the diameter (d) of any circle and divide — you always get the same number, a little more than 3. That ratio is π (pi):

$\pi =\frac{C}{d}\approx 3.14159\left(\approx \frac{22}{7}\right)$

So the circumference is:

$C=\pi d=2\pi r$

3. Why π is irrational

π's decimal expansion is non-terminating and non-recurring — it cannot be written exactly as any fraction. 22/7 and 3.14 are only approximations. (This was proved rigorously by Lambert in 1761.)

4. Length of an arc and perimeter of a sector

An arc is part of the circumference. For a central angle θ (in degrees), the arc length is the matching fraction of the full circumference:

$\text{arc length}=\frac{\theta }{360^{\circ }}\times 2\pi r$

The perimeter of a sector = arc length + the two radii:

$P_{\text{sector}}=\frac{\theta }{360^{\circ }}\times 2\pi r+2r$

5. Areas

• Circle: $A=\pi r^2$
• Sector (angle θ): $A=\frac{\theta }{360^{\circ }}\times \pi r^2$
• Square: $a^2$; Rectangle: $l\times b$; Triangle: $\frac{1}{2}\times \text{base}\times \text{height}$

6. Worked examples

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

$C=2\pi r\Rightarrow 44=2\times \frac{22}{7}\times r\Rightarrow r=\frac{44\times 7}{2\times 22}=\frac{308}{44}=$ 7 cm.

(b) Find the area of a circle of radius 7 cm.

$A=\pi r^2=\frac{22}{7}\times 7\times 7=22\times 7=$ 154 cm².

(c) Find the length of an arc subtending 90° in a circle of radius 14 cm.

$\text{arc}=\frac{90}{360}\times 2\times \frac{22}{7}\times 14=\frac{1}{4}\times 88=$ 22 cm.

7. Quick recap

• Perimeter = boundary length (add the sides); area is in square units.
• π = C/d (same for every circle), ≈ 3.14159 ≈ 22/7, and irrational (22/7, 3.14 are approximations).
• Circumference $C=2\pi r=\pi d$; area $A=\pi r^2$.
• Arc length $=\frac{\theta }{360}\times 2\pi r$; sector perimeter = arc + 2r; sector area $=\frac{\theta }{360}\times \pi r^2$.
• Choose π = 22/7 when the radius is a multiple of 7; otherwise 3.14.