Three equal circles fit exactly inside the rectangle, touching each other and the sides of the rectangle. Class 9

Three equal circles fit exactly inside the rectangle, touching each other and the sides of the rectangle. Class 9

Question 1.

Solution:

Three equal circles fit exactly inside the rectangle, touching each other and the sides of the rectangle.

Let radius of each circle = r

∵ Height of rectangle = diameter = 2r

∴ Length of rectangle = 3 × 2r = 6r

Area of rectangle = Length × breadth

= 6r × 2r = 12r²

Area of one circle = πr²

∴ Area of 3 circles = 3πr²

Fraction of rectangle covered with circles Area of circles Area of rectangle

$\begin{aligned} & =\frac{\text { Area of circles }}{\text { Area of rectangle }} \\ & =\frac{3 \pi r^2}{12 r^2} \\ & =\frac{\pi}{4}\end{aligned}$

∴ Required fraction = $\frac{\pi}{4}$

Four equal circles fit exactly inside the rectangle, touching each other and sides of the rectangle.

Let radius of each circle = r

∵ Height of rectangle = diameter = 2r

∴ Length of rectangle = 4 × 2r = 8r

Area of rectangle = Length × Breadth

= 8r × 2r = 16r²

∵ Area of one circle = πr²

∴ Area of 4 circles = 4πr²

Fraction of rectangle covered with circles

$\begin{aligned} & =\frac{\text { Area of circles }}{\text { Area of rectangle }} \\ & =\frac{4 \pi r^2}{16 r^2} \\ & =\frac{\pi}{4}\end{aligned}$

∴ Required fraction = $\frac{\pi}{4}$