a square has an area of 100 cm^2. how much bigger is the square's circumscribed circle than the inscribed circle?

The circumscribed circle would have a diameter equal to the diagonal of the square

each side of the square is 10 cm
using Pythagoras, the diagonal has a length of 10√2
so the radius is 5√2

area of circle = π(5√2)^2 = 50π
area of square = 100

difference = (50π - 100) cm^2 , (appr 57.1)

To find the difference in size between the square's circumscribed circle and its inscribed circle, we first need to calculate the radius of each circle.

Let's start with the inscribed circle:

In a square, the diagonals bisect each other at right angles, therefore, the diagonal of the square is equal to the diameter of its inscribed circle.

Since the area of the square is given as 100 cm^2, we know that the side length of the square can be calculated by taking the square root of the area.

Side length of the square = √100 cm = 10 cm.

The diagonal of the square (diameter of the inscribed circle) can be found using Pythagoras' theorem:

Diagonal of the square = √(side length^2 + side length^2) = √(10^2 + 10^2) = √200 cm.

Next, let's find the radius of the inscribed circle:

Radius of the inscribed circle = (diagonal of the square) / 2 = (√200 cm) / 2 = √50 cm = 5√2 cm.

Now, let's move on to the circumscribed circle:

The diameter of the circumscribed circle is equal to the diagonal of the square. We already found the diagonal to be √200 cm.

Therefore, the radius of the circumscribed circle is:
Radius of the circumscribed circle = (diagonal of the square) / 2 = (√200 cm) / 2 = √50 cm = 5√2 cm.

To find the difference in size between the circumscribed and inscribed circles, we need to subtract the radius of the inscribed circle from the radius of the circumscribed circle:

Difference = (Radius of the circumscribed circle) - (Radius of the inscribed circle).
Difference = (5√2 cm) - (5√2 cm) = 0 cm.

Hence, the circumscribed circle is the same size as the inscribed circle in this case.