# Math Help - Equilateral shaped dartboard has a circle in it. probability of dart hitting the circ

1. ## Equilateral shaped dartboard has a circle in it. probability of dart hitting the circ

This is an IGCSE probability question that is giving me a lot of trouble. Since it's not just a probability problem and that the non probability part is the part giving me trouble I decided to put it here.

A dartboard is in the shape of an equilateral triangle inside which is inscribed a circle.
A dart is randomly thrown at the board (Assume that it hits the board).

A) Given that tan 60 degrees = \sqrt{3} and sin 60 degrees =
\sqrt{3} /2

show that the probability of the dart hitting the board inside the circle is
\pi /3 \sqrt{3}

The diagram given looks like this: http://tinyurl.com/3nym92q

My working:
The probability should be the area of the circle (\pi r^2) over the area of the triangle(half* unknown base * unknown height)

So you should somehow end up with \pi r^2/ 3r^2 \sqrt{3} which simplifies down to \pi /3 \sqrt{3}

But how do use the information in the question to find out that the area of the triangle is 3r^2\sqrt{3} ?

2. Find the center of the circle.
From this center, draw the three radii perpendicular to the three sides of the triangle.
From this center, draw the three line segments to the vertices of the triangle.

There's a whole lot of information in that structure.

3. Originally Posted by Paul82
A dartboard is in the shape of an equilateral triangle inside which is inscribed a circle. A dart is randomly thrown at the board (Assume that it hits the board).
Suppose that the length of a side of the equilateral triangle is $\mathbf{s}.$
Then the area is $\mathcal{A}=\frac{\sqrt{3}\mathbf{s}^2}{4}$.

The radius of the inscribed circle is $\mathbf{r}=\frac{\sqrt{3}\mathbf{s}}{6}$.