Question

15 the figure below shows three small circles, each with a diameter of 6 centimeters, inside a larger circle.
what is the area of the shaded part of the figure?
a square centimeters
b 81π square centimeters
c square centimeters
d square centimeters

Explanation:

Step1: Calculate the radius of small circles

The diameter of each small circle \(d = 6\space cm\), so the radius \(r=\frac{d}{2}=3\space cm\)

Step2: Calculate the radius of the large circle

The diameter of the large circle is equal to the sum of the diameters of three small circles. So \(D=6 + 6+6=18\space cm\), and the radius \(R=\frac{D}{2}=9\space cm\)

Step3: Calculate the area of the large circle

The area formula of a circle is \(A=\pi R^{2}\). Substitute \(R = 9\) into the formula, we get \(A_{large}=\pi\times9^{2}=81\pi\space cm^{2}\)

Step4: Calculate the area of three small circles

The area of one small circle \(a=\pi r^{2}\). Substitute \(r = 3\) into the formula, \(a=\pi\times3^{2}=9\pi\space cm^{2}\). The area of three small circles \(A_{small}=3\times9\pi = 27\pi\space cm^{2}\)

Step5: Calculate the area of the shaded part

The area of the shaded part \(A=A_{large}-A_{small}\). Substitute \(A_{large}=81\pi\) and \(A_{small}=27\pi\) into the formula, \(A=(81 - 27)\pi=54\pi\space cm^{2}\)

Answer:

B. \(54\pi\) square centimeters