5. find the ratio between the area of a square inscribed in a circle and an equilateral circumscribed about the same circle.
make a good sketch,
let each side of the equilateral triangle be 2
then the height of the equilateral triangle ....
using the 30-60-90 triangle ratios, is √3
area of triangle = (1/2)(2)√3 = √3
You should have a small equilateral triangle sitting on top of the square with sides 1
so the side of the square is 1
and its area is 1x1 = 1
ration of square : triangle = 1 : √3
= 1/√3 = √3/3 or √3 : 3
why did u include triangle ? wait i dot understand
To find the ratio between the area of a square inscribed in a circle and an equilateral triangle circumscribed about the same circle, we can use the following steps:
1. Let's denote the radius of the circle as R.
2. The diagonal of the square inscribed in the circle is equal to the diameter of the circle, which is 2R. Since the diagonals of a square are equal in length, each side length of the square is 2R / √2 = R√2.
3. The area of the square is calculated as the square of the side length, so the area of the square is (R√2)² = 2R².
4. The length of each side of the equilateral triangle is equal to the diameter of the circle, which is 2R.
5. The height of an equilateral triangle is found by dividing the length of one side by 2 and multiplying by √3, so the height of this equilateral triangle is (2R / 2) * √3 = R√3.
6. The area of an equilateral triangle is found by dividing the product of the base and height by 2, so the area of this equilateral triangle is (2R * R√3) / 2 = R²√3.
7. Finally, to find the ratio between the area of the square and the area of the equilateral triangle, we divide the area of the square by the area of the equilateral triangle: (2R²) / (R²√3) = 2 / √3.
Therefore, the ratio between the area of a square inscribed in a circle and an equilateral triangle circumscribed about the same circle is 2/√3.
To find the ratio between the area of a square inscribed in a circle and an equilateral triangle circumscribed about the same circle, we can follow these steps:
1. Let's start by finding the area of the square inscribed in the circle.
a. The square is inscribed in the circle, which means its diagonal is equal to the diameter of the circle.
b. Let's assume the side length of the square is "s". Then the diagonal of the square would be "√2s" (using Pythagorean theorem).
c. Since the diagonal of the square is equal to the diameter of the circle, we have: √2s = 2r, where "r" is the radius of the circle.
d. Solving for "s", we get: s = r/√2.
e. The area of the square is calculated by multiplying the side length with itself: Area_square = s^2 = (r/√2)^2 = r^2/2.
2. Now let's find the area of the equilateral triangle circumscribed about the circle.
a. The lengths of the sides of the equilateral triangle are equal to the diameter of the circle.
b. Therefore, each side length of the triangle is equal to 2r.
c. The height of an equilateral triangle can be found by dividing the length of one side by 2 and multiplying it by the square root of 3: Height_triangle = (2r)/2 * √3 = r√3.
d. The area of an equilateral triangle is calculated by multiplying the base (side length) with the height and dividing the result by 2: Area_triangle = (2r * r√3)/2 = r^2√3.
3. Finally, to find the ratio between the area of the square and the equilateral triangle, divide the area of the square by the area of the triangle:
Ratio = Area_square / Area_triangle = (r^2/2) / (r^2√3) = 1/(2√3) ≈ 0.2887.
Therefore, the ratio between the area of a square inscribed in a circle and an equilateral triangle circumscribed about the same circle is approximately 0.2887.