For the triangle below, let x be the area of the circumcircle, and let y be the area of the incircle. Compute x - y.
We can compute the areas of the circumcircle and incircle of an equilateral triangle with side length "a" using formulas involving their radii.
First, let's find the circumradius (R) and the inradius (r) of the equilateral triangle.
For an equilateral triangle with side length "a", the circumradius R is given by the formula:
R = a / (2 * sin(π / 3))
Since sin(π / 3) = √3 / 2, we have:
R = a / (2 * (√3 / 2))
R = a / √3
Now, let's find the inradius. In an equilateral triangle, the inradius r is given by the formula:
r = a / (2 * √3)
Now that we have the radii, we can compute the areas of the circumcircle (x) and the incircle (y) using the formulas:
x = π * R^2
y = π * r^2
Substituting the expressions for R and r, we have:
x = π * (a / √3)^2
x = π * (a^2 / 3)
y = π * (a / (2 * √3))^2
y = π * (a^2 / 12)
Now, compute x - y:
x - y = π * (a^2 / 3) - π * (a^2 / 12)
x - y = (4 * π * a^2 / 12) - (π * a^2 / 12)
x - y = (3 * π * a^2 / 12)
x - y = (π * a^2 / 4)
So the difference between the areas of the circumcircle and the incircle of an equilateral triangle with side length "a" is (π * a^2 / 4).
