Chapter 11, Problem 42RP

To determine

To calculate:Theshaded area if all 12 sides of the cross are congruent.

Answer to Problem 42RP

The shadedarea is $45.66$ .

Explanation of Solution

Given information:

All 12 sides of the cross are congruenteach having a length of 4.

Formula used:

Area of square: $A=s^2$

s = side of square

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

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

r = radius of circle.

Calculation:

  

  $\begin{array}{l}{A}_{Square}=s^2\\ A_{Square}={\left(4\right)}^2\\ A_{Square}=16\\ A_{Cross}=5\ast A_{Square}\\ A_{Cross}=5\ast 16\\ A_{Cross}=80\end{array}$

To find radius of circle we need to calculate $\overline{AB}$ .

In right angled triangle ABC , we get

  $\begin{array}{l}{\left(AB\right)}^2={\left(AC\right)}^2+{\left(BC\right)}^2\\ {\left(AB\right)}^2={\left(12\right)}^2+{\left(4\right)}^2\\ {\left(AB\right)}^2=144+16\\ {\left(AB\right)}^2=160\\ AB=12.64\\ r=\frac{AB}{2}=\frac{12.64}{2}=6.32\end{array}$

  $\begin{array}{l}{A}_{Circle}=\pi r^2\\ A_{Circle}=\pi {\left(6.32\right)}^2\\ A_{Circle}=40\pi \end{array}$

Shaded Area = Area of circle - Area of cross

Shaded Area $=80-40\pi$

Shaded Area $=45.66$