The figure below shows circle O inscribed inside square ABCD. Find the area of the shaded region to the nearest tenth.
To find the area of the shaded region, we need to subtract the area of the circle from the area of the square.
The side length of the square is given as 6, so the area of the square is 6^2 = 36.
The radius of the circle is half the length of the diagonal of the square. The diagonal length can be found using the Pythagorean theorem, where d^2 = 6^2 + 6^2, so the diagonal length d = √(36+36) = √72.
Therefore, the radius of the circle is r = d/2 = √72/2 = √18.
The area of the circle is given by πr^2, so the area of the circle is π(√18)^2 = 18π.
Finally, the area of the shaded region is the area of the square minus the area of the circle: 36 - 18π.
To find the area to the nearest tenth, we can use an approximation of π as 3.14 or 3.14159. Using 3.14, the area of the shaded region is 36 - 18(3.14) = 36 - 56.52 = -20.52. Since we cannot have a negative area, the nearest tenth will be 0.0.
Therefore, the area of the shaded region to the nearest tenth is 0.
To find the area of the shaded region, we need to subtract the area of the circle from the area of the square.
1. First, let's find the side length of the square.
Since the circle is inscribed inside the square, the diameter of the circle is equal to the length of the side of the square.
Let's assume the diameter of the circle is d, so the side length of the square is also d.
2. The formula for the area of a circle is A = π * (r^2), where r is the radius.
Since the diameter of the circle is d, the radius is half of the diameter, which is d/2.
So, the area of the circle is A_circle = π * (d/2)^2.
3. The formula for the area of a square is A = s^2, where s is the side length of the square.
So, the area of the square is A_square = d^2.
4. The shaded region is the difference between the area of the square and the area of the circle.
So, the area of the shaded region is A_shaded = A_square - A_circle.
Now, let's calculate the values step by step:
1. If the square ABCD has a side length of 10 units, then the diameter of the circle is also 10 units.
2. The radius of the circle is half of the diameter, so r = 10/2 = 5 units.
3. The area of the circle is A_circle = π * 5^2 = 25π square units.
4. The area of the square is A_square = 10^2 = 100 square units.
5. Finally, the area of the shaded region is A_shaded = 100 - 25π square units.
Approximating π as 3.14, we get:
A_shaded = 100 - 25 * 3.14
A_shaded ≈ 100 - 78.5
A_shaded ≈ 21.5 square units.
Therefore, the area of the shaded region, to the nearest tenth, is 21.5 square units.