FORM GR1268 Question 2

Question:
What is the area of an equilateral triangle whose inscribed circle has radius 2?

(A) 12     (B) 16     (C) \(12\sqrt3\)     (D) \(16\sqrt3\)     (E) \(4(3+2\sqrt2)\)

Answer:
(C)

Answer Key:

An equilateral triangle contains six right angled triangles as shown above.
Each of the right angled triangle has \(90^\circ\)-\(30^\circ\)-\(60^\circ\) angles.
The ratio of the corresponding sides is \(2:1:\sqrt3\).

Since we know the radius of the circle is 2, we can calculate the length of the remaining two sides.
Finally, \(\displaystyle A=\frac{bh}{2}=\frac{4\sqrt3\cdot6}{2}=12\sqrt3\).

If you don't remember the ratio of the sides for \(90^\circ\)-\(30^\circ\)-\(60^\circ\) triangle,
start with an equilateral triangle with side length 2, and cut it into half.
Then use the Pythagorean theorem: \(1^2+s^2=2^2\) and \(s=\sqrt{2^2-1^2}=\sqrt3\).