To find the area of a circle that has a square inscribed within it, you first need to understand the relationship between the dimensions of the circle and the square.
Let’s break it down:
- Understand the geometry: An inscribed square means that all four corners of the square touch the circle. If the side length of the square is ‘s’, then the diagonal of the square will equal the diameter of the circle.
- Calculate the diagonal of the square: The formula for the diagonal (d) of a square in terms of its side length (s) is:
d = s√2. - Relate the diagonal to the diameter of the circle: Since the diagonal of the square is equal to the diameter of the circle, we can say: d = D = s√2, where D is the diameter of the circle.
- Find the radius of the circle: The radius (r) of the circle is half of the diameter, so r = D/2 = (s√2)/2.
- Calculate the area of the circle: The area (A) of the circle can be found using the formula: A = πr². Replacing r with our expression gives:
A = π((s√2)/2)² = π(s²/2).
So, the final formula for the area of the circle in terms of the side length of the inscribed square is:
A = (π/2) * s²
This formula allows you to calculate the area of the circle whenever you know the side length of the square inside it.