When a square is inscribed within a circle, you can easily find one shape's area from the other one's. The radius of the circle, which determines its area, is half the length of the square's diagonal. The length of this diagonal forms a right-angled triangle with the length and width of the square. This means that you can calculate the diagonal's length using the Pythagorean theorem, which relates the lengths of a right-angled triangle's sides.
Find the square root of the square's area. For instance, if the square has an area of 100 in²: √100 = 10 in. This is the length of each of the square's sides.
Square this length again, and multiply the result by 2: 2 × 10² = 200. This is the sum of the sides' squared lengths.
Find the square root of this answer: √200 = 14.14. This is the length of the square's diagonal.
Divide the result by 2: 14.14 ÷ 2 = 7.07. This is the length of the circle's radius.
Square the radius, and multiply the result by the constant pi: 7.07² × 3.142 = 157 in². This is the circle's area.
To convert in a single step, simply multiply the square's area by 1.571, which is half of pi.