Question

the length of the square and the diameters of the circles are all 4 units. find the combined area of the shaded regions. the combined area of the shaded regions is . (simplify your answer. type an exact answer, using π as needed.)

Explanation:

Step1: Calculate area of the square

The side - length of the square is equal to the diameter of the circle, $d = 4$ units. The area of a square formula is $A_{square}=s^{2}$, so $A_{square}=4^{2}=16$ square units.

Step2: Calculate area of the four quarter - circles in the square

The four quarter - circles in the square together form a complete circle. The radius of the circle $r=\frac{d}{2}=\frac{4}{2} = 2$ units. The area of a circle formula is $A_{circle}=\pi r^{2}$, so $A_{circle}=\pi\times2^{2}=4\pi$ square units. The area of the shaded part in the square is $A_{1}=A_{square}-A_{circle}=16 - 4\pi$ square units.

Step3: Calculate area of the two circles

The radius of each circle $r = 2$ units. The area of one circle is $A_{circle}=\pi r^{2}=4\pi$ square units. The area of two circles is $A_{circles}=2\times4\pi = 8\pi$ square units.

Step4: Calculate area of the rectangle enclosing the two circles

The length of the rectangle is equal to the sum of the diameters of the two circles, $l=4 + 4=8$ units, and the width is equal to the diameter of one circle, $w = 4$ units. The area of the rectangle is $A_{rectangle}=l\times w=8\times4 = 32$ square units. The area of the shaded part in the rectangle is $A_{2}=A_{rectangle}-A_{circles}=32-8\pi$ square units.

Step5: Calculate the combined shaded area

The combined shaded area $A=A_{1}+A_{2}=(16 - 4\pi)+(32-8\pi)=48 - 12\pi$ square units.

Answer:

$48 - 12\pi$