A circle of radius 7\text{ cm} is inscribed inside a square. What is the area of the regio… — JAMB UTME

A circle of radius $7\text{ cm}$ is inscribed inside a square. What is the area of the region inside the square but outside the circle? (Take $\pi = \frac{22}{7}$ )

$42\text{ cm}^2$ Answer
B
$154\text{ cm}^2$
C
$196\text{ cm}^2$
D
$12\text{ cm}^2$

Answer

The area of the region inside the square but outside the circle is

42\text{ cm}^2

Because the circle is inscribed inside the square, the diameter of the circle is equal to the side length of the square:

s = 2r = 2(7) = 14\text{ cm}

. The area of the square is

14^2 = 196\text{ cm}^2

. The area of the circle is

\pi r^2 = \frac{22}{7} \times 7^2 = 154\text{ cm}^2

. Subtracting the area of the circle from the area of the square gives the region inside the square but outside the circle:

196 - 154 = 42\text{ cm}^2

Step-by-Step Solution

Determine the side length of the square.

Side length

s = 2 \times 7\text{ cm} = 14\text{ cm}

An inscribed circle touches all four sides of the square, so its diameter equals the side length of the square.

Calculate the area of the square.

Area of square

= s^2 = 14^2 = 196\text{ cm}^2

The area of a square with side length

s

is given by

s^2

Calculate the area of the inscribed circle.

Area of circle

= \pi r^2 = \frac{22}{7} \times 7^2 = 22 \times 7 = 154\text{ cm}^2

The area of a circle with radius

r

is given by

\pi r^2

Subtract the circle's area from the square's area to find the area of the shaded outer region.

Area

= 196\text{ cm}^2 - 154\text{ cm}^2 = 42\text{ cm}^2

The remaining area consists of the four corner regions between the circle and the bounding square.

Key Concept

Area of composite plane figures (shaded area between inscribed shape and container)

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