To find the area of a square inscribed in a circle, you first need to understand the relationship between the square and the circle.
1. **Identify the Circle’s Radius**: Start with the radius of the circle, which we will denote as ‘r’. This radius is the distance from the center of the circle to any point on its boundary.
2. **Relationship Between the Square and Circle**: In an inscribed square, the corners of the square touch the circle. The diagonal of the square is equal to the diameter of the circle. Since the diameter is twice the radius, we can express this as:
Diagonal of square = 2r
3. **Calculate the Side Length of the Square**: The diagonal ‘d’ of a square with side length ‘s’ can be calculated using the formula:
d = s√2
Setting the two diagonal equations equal gives:
2r = s√2
4. **Solve for the Side Length**: Rearranging the equation to solve for ‘s’ gives:
s = (2r) / √2 = r√2
5. **Find the Area of the Square**: The area ‘A’ of the square can be calculated using the formula:
A = s²
Plugging in our value for ‘s’, we get:
A = (r√2)² = 2r²
So, the area of a square that is inscribed in a circle with radius ‘r’ is 2r².