Question

An equilateral triangle with side of 2 square root 3 is inscribed in a circle. What is the area of one of the sectors formed by the radii to the vertices of the triangle? 1.33 sq. in. 2 sq. in. 2.09 sq. in.

Answer 1

Refer to the figure.

We are looking for the area of the sector of a circle as shown in the figure shaded with green color. 

The area of a sector of a circle can be calculated using the formula $A=\frac{1}{2}r^2sin\left(\theta \right)$
     where r=radius, and θ=central angle (in radians)

The central angle of the given sector is just one-third of a full circle (2π). That is 
     $\theta =\frac{2\pi }{3}$

Now, to solve for the radius of the circle, we can use the formula
     $R=\frac{abc}{4A}$
          where R is the radius of the circumscribed circle; a,b, and c are the sides of the triangle; and A is the area of the triangle.

The area of the equilateral triangle can be solved using the formula $A=\frac{\sqrt{3}}{4}a^2$. That is 
     $A=\frac{\sqrt{3}}{4}a^2=\frac{\sqrt{3}}{4}\left(2\sqrt{3}\right)^2=3\sqrt{3}$

Now, we substitute this area in the formula to solve for the radius of the circle. 
     $R=\frac{abc}{4A}=\frac{\left(2\sqrt{3}\right)^3}{4\left(3\sqrt{3}\right)}=2$

Finally, we can solve for the area of the sector by substituting the values of the angle θ, and the radius.
     $A=\frac{1}{2}r^2\theta =\frac{1}{2}\left(2\right)^2\left(\frac{2\pi }{3}\right)=\frac{4\pi }{3}\:square\:units$

Answer 2

Answer:

1.33

Step-by-step explanation: