Question

The circumference of a circle circumscribing an equilateral triangle is \[24\pi \] units. The area of the circle inscribed in the equilateral triangle is \[4a\pi \]. Find \[a\].

Answer 1

Hint: We will first draw the figure following the statement given in the question. As we are given that the circumference of a circle will be equal to \[24\pi \] units. So, from here we can find the radius of the circumcircle. As the circle is inscribed in an equilateral triangle, we will determine the radius of the inner circle using the figure. Thus, we will find the area of the inner circle using the formula of area of the circle, \[\pi {r^2}\], and then we will compare it with \[4a\pi \] to find the value of \[a\].

Complete step by step answer:

We are given that the circumference of the circle which is inscribing an equilateral triangle is \[24\pi \] units.
We will first construct the figure using the given data.

As given in the question that the circumference of the circle is equal to \[24\pi \] units.
Thus, we get,
\[
   \Rightarrow 2\pi R = 24\pi \\
   \Rightarrow R = 12 \\
\]
As we can see from the figure that we can use the trigonometric identity that \[\sin \left( \alpha \right) = \dfrac{P}{H}\] to determine the value of the radius of the inner circle which is inscribed in the equilateral triangle.
And the equilateral triangle has 60 degrees each so, we can use the value of \[\alpha = 30\].
The perpendicular value represents the radius of the inner circle and the hypotenuse represents the radius of the outer circle.
Thus, we have,
\[
   \Rightarrow \sin \left( {30} \right) = \dfrac{r}{R} \\
   \Rightarrow \dfrac{1}{2} = \dfrac{r}{{12}} \\
   \Rightarrow r = 6 \\
\]
Now, we can find the area of inner circle by substituting the value of radius in the formula \[A = \pi {r^2}\]
Thus, we have,
\[
   \Rightarrow A = \pi {\left( 6 \right)^2} \\
   \Rightarrow A = 36\pi \\
\]
Now, we will determine the value of \[a\] by comparing the obtained area of incircle by \[4a\pi \],
Hence, we have,
\[
   \Rightarrow 4a\pi = 36\pi \\
   \Rightarrow a = 9 \\
\]
Hence, we can conclude that the value of \[a = 9\].

Note: Construct the figure properly and do not forget to represent the angle of the equilateral triangle. As the equilateral triangle makes the angle of \[60^\circ \] each so as we have made the triangle in the inner circle with one of its vertices as the vertex of the equilateral triangle so we get the angle as \[30^\circ \]. Do remember the formula for the area of circle and circumference of the circle which is required in the question.