5. find the ratio between the area of a square inscribed in a circle and an equilateral circumscribed about the same circle.
To find the ratio between the area of a square inscribed in a circle and an equilateral triangle circumscribed about the same circle, we need to understand the properties of these shapes.
Let's start with the square inscribed in the circle.
1. Draw a square inside a circle such that its four corners touch the circle.
2. The diagonal of the square is equal to the diameter of the circle. Let's call this diagonal "d".
Next, let's focus on the equilateral triangle circumscribed about the same circle.
1. Draw an equilateral triangle outside the circle such that all three vertices touch the circle.
2. The side length of the equilateral triangle is equal to the diameter of the circle. Let's call this side length "s".
Now, let's find the area of these shapes.
For the square:
- The area of a square is given by the formula A = side^2.
- Since the diagonal of the square is equal to the diameter of the circle, we can write the side length in terms of the diameter: s = d.
So, the area of the square is A_square = s^2 = d^2.
For the equilateral triangle:
- The area of an equilateral triangle is given by the formula A = (sqrt(3)/4) * side^2.
- Since the side length of the equilateral triangle is equal to the diameter of the circle, we can write the side length in terms of the diameter: s = d.
So, the area of the equilateral triangle is A_triangle = (sqrt(3)/4) * s^2 = (sqrt(3)/4) * d^2.
To find the ratio between the two areas, we divide the area of the square by the area of the triangle:
Ratio = A_square / A_triangle = (d^2) / [(sqrt(3)/4) * (d^2)] = 4 / sqrt(3).
Therefore, the ratio between the area of a square inscribed in a circle and an equilateral triangle circumscribed about the same circle is 4 / sqrt(3).