Question

The smaller circles are congruent. Find the area of the shaded region. Round your answer to the nearest hundredth.

Answer 1

56.552 m²

Explanation:

Recall the equation for the area of a circle;

A = πr²

r of smaller circles = 3m

r of larger circle = 3m x 2 = 6m

Area of each smaller circle: 28.274 m²

Area of the larger circle: 113.1 m²

To find the area of the shaded region, take the area of the larger circle - the area of the smaller circle(s).

113.1 m² - 2(28.274 m²) = 56.552 m²

Answer 2

The area of the shaded region is 56.55 m² rounded to the nearest hundredth.

Explanation:

From inspection of the given diagram, the radius of the two congruent unshaded circles is 3 m.

As the two congruent unshaded circles touch each other and touch the circumference of the larger circle, the radius of the larger circle is the diameter of one of the unshaded circles, i.e. 6 m.

To calculate the area of the shaded region, subtract the areas of the congruent unshaded circles from the area of the larger circle.

The formula for the area of a circle is πr², where r is the radius.

Therefore:

`\begin{aligned}\textsf{Area of shaded region}&=\textsf{Area of larger circle}-2(\textsf{Area of unshaded circle})\\&=\pi \cdot 6^2-2(\pi \cdot 3^2)\\&=36\pi-2(9\pi)\\&=36\pi-18\pi\\&=18\pi\\&=56.5486677...\\&=56.55\;\sf m^2\;(nearest\;hundredth)\end{aligned}`

Therefore, the area of the shaded region is 56.55 m² rounded to the nearest hundredth.