Answer:8π
Explanation:
To find the area of the circle circumscribed about an isosceles right triangle, we first need to determine the length of the hypotenuse of the triangle.
Given that the area of the isosceles right triangle is 8, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Since the triangle is isosceles and right-angled, the base and height are equal. Let's denote the base and height as 'x'.
8 = (1/2) * x * x
16 = x^2
x = √16
x = 4
Therefore, the length of the hypotenuse of the triangle is 4√2 (using the Pythagorean theorem).
Now, the diameter of the circle is equal to the hypotenuse of the triangle. So, the diameter of the circle is 4√2.
The radius of the circle is half the diameter, so the radius is (4√2)/2 = 2√2.
The area of a circle is calculated using the formula:
Area = π * radius^2
Plugging in the value of the radius, we get:
Area = π * (2√2)^2
Area = π * 4 * 2
Area = 8π
Therefore, the area of the circle circumscribed about the isosceles right triangle with area 8 is 8π.