Question

A circle is inscribed in an equilateral triangle. What is the probability that a point chosen at random in the triangle will be inside the circle?

Answer 1

π/(3√3)

Explanation:

The probability = area of circle/ area of triangle

Assuming the triangle has a unit lengths, it's area= 1/2bh = 1/2x1x√3/2=√3/4

Note: the height,h =√(1²-(1/2)²)=√3/4=√3/2

Area of circle = πr²= π(1/2√3)²=π/12

Note: To get the radius of the circle

(√3/2-r)²=r²+(1/2)²

3/4+r²-r√3=r²+1/4

3/4-1/4=r√3

1/2=r√3

r= 1/2√3

Prob = area of circle/ area of triangle=(π/12)/(√3/4)=π/(3√3)