Given a circle with an inscribed equilateral triangle with each side of the triangle having a measurement of 18cm. What is the probability of selecting a point at random inside the triangle, assuming that the point cannot lie outside the circular region? Leave your answer in exact form.
probablity= area triangle/area circle
To find the probability of selecting a point at random inside the triangle, we need to compare the areas of the triangle and the circular region.
Let's start by finding the area of the equilateral triangle.
The formula to calculate the area of an equilateral triangle is:
Area = (sqrt(3) / 4) * (side^2)
In this case, the side of the triangle is given as 18 cm, so the area of the triangle is:
Area of triangle = (sqrt(3) / 4) * (18^2)
= (sqrt(3) / 4) * 324
= 81√3 cm²
Now, let's find the radius of the circle.
In an equilateral triangle, the radius of the circumcircle is equal to one-third of the length of one of its sides.
So, the radius of the circle inscribed in the triangle is 18/3 = 6 cm.
Now, we can find the area of the circular region:
Area of circular region = π * (radius^2)
= π * (6^2)
= 36π cm²
To find the probability, we compare the area of the triangle to the area of the circular region:
Probability = Area of triangle / Area of circular region
= (81√3 cm²) / (36π cm²)
So, the probability of selecting a point at random inside the triangle is (81√3 / 36π), which is the exact form of the probability.