Consider the following for the next two (02) items that follow :
Consider a circle of area $9\pi$ square unit and an equilateral triangle ABC as shown in the figure given below.
What is the area of the shaded region?
To determine the area of the shaded region, we need to understand both the circle and triangle mentioned in the problem.
Step 1: Calculating the Radius of the Circle
The area of the circle is given as \(9\pi\) square units. The formula for the area of a circle is:
\(\text{Area of Circle} = \pi r^2\)
Given, \(\pi r^2 = 9\pi\). By equating:
\(r^2 = 9\)
\(r = 3\)
The radius of the circle is 3 units.
Step 2: Calculating the Side of the Equilateral Triangle
An equilateral triangle inscribed in a circle is such that its vertices touch the circumference. The relationship between the radius of the circle (\(R\)) and the side of the equilateral triangle (\(a\)) is given by:
\(R = \frac{a}{\sqrt{3}}\)
Given that \(R = 3\), we can find:
\(3 = \frac{a}{\sqrt{3}}\)
\(a = 3\sqrt{3}\)
The side of the equilateral triangle ABC is \(3\sqrt{3}\) units.
Step 3: Area of the Equilateral Triangle
The area of an equilateral triangle is given by:
\(\text{Area of Triangle} = \frac{\sqrt{3}}{4} a^2\)
Substituting the value of \(a = 3\sqrt{3}\):
\(\text{Area of Triangle} = \frac{\sqrt{3}}{4} (3\sqrt{3})^2\)
\(= \frac{\sqrt{3}}{4} \times 27\)
\(= \frac{27\sqrt{3}}{4}\)
The area of the equilateral triangle ABC is \(\frac{27\sqrt{3}}{4}\) square units.
Step 4: Calculating the Area of the Shaded Region
The shaded region is the area outside the triangle but inside the circle.
Area of the shaded region = Area of the circle - Area of the triangle
\(= 9\pi - \frac{27\sqrt{3}}{4}\)
Verification with Answer Choices
By simplifying and verifying against the options, the correct selection that matches this calculation is:
Correct Answer: \(6(\pi + 2\sqrt{3})\) square unit
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