Chapter 11, Problem 35RP

To determine

To calculate:Theshaded area if each small circle has a radius of $3$ .

Answer to Problem 35RP

The shadedarea is $51.71$ .

Explanation of Solution

Given information:

Five circles are tangent.

Each small circle has a radius of $3$ .

Formula used:

Area of a circle: $A=\pi r^2$

r = radius of circle.

Calculation:

  

To find the radius of longer circle, draw a square to connect the centers of the four small circles.Then connect the center of longer circle with the center of two smaller circles to form triangle. Legs are congruent because they are difference between radius of larger circle and smaller circle.

Right angle and ${45}^{\circ }{45}^{\circ }{90}^{\circ }\Delta$ is formed.

Hypotenuse = 6, each leg $=3\sqrt{2}$

Radius of smaller circle = 3,

Radius of larger circle $=3\sqrt{2}+3$

Area of longer circle:

  $\begin{array}{l}{A}_{Longercircle}=\pi {\left(3+3\sqrt{2}\right)}^2\\ A_{Longercircle}=\pi (9+18\sqrt{2}+18)\\ A_{Longercircle}=27\pi +18\sqrt{2}\\ A_{Longercircle}=164.79\end{array}$

Area of smaller circle:

  $\begin{array}{l}{A}_{Smallercircle}=\pi {\left(3\right)}^2\\ A_{Smallercircle}=\pi (9)\\ A_{Smallercircle}=9\pi \\ A_{Smallercircle}=28.27\end{array}$

Shaded Area = Area of longer circle - 4( Area of Smaller circle)

Shaded Area $=164.79-\left(4\ast 28.27\right)=164.79-113.08=51.71$