Calculate the perimeter of the inscribed square given that the radius of the circle is 3 feet.
draw a diagram. The diagonal of the square is the diameter of the circle.
so, since that's 6 feet, the side of the square is 6/√2 = 3√2. The perimeter is thus 12√2
To calculate the perimeter of the inscribed square, we need to find the length of one side of the square. We can do this by using the geometry concept that states the diagonal of a square is equal to the diameter of the inscribed circle.
Step 1: Find the diameter of the circle
Given the radius of the circle is 3 feet, we know that the diameter is twice the radius. So, the diameter is 2 * 3 = 6 feet.
Step 2: Find the length of one side of the square
Since the diagonal of the square is equal to the diameter of the circle, and the diagonal of a square divides it into two congruent right triangles, we can use the Pythagorean Theorem to find the length of one side.
Let's call the length of one side of the square 's'. Then, using the Pythagorean Theorem, we have:
s^2 + s^2 = (6)^2
2s^2 = 36
s^2 = 36/2
s^2 = 18
s = sqrt(18)
s ≈ 4.24 feet (rounded to two decimal places)
Step 3: Calculate the perimeter of the square
The perimeter of a square is found by adding together all four sides. Since the square has all sides equal, we can simply multiply the length of one side by 4.
Perimeter = 4 * s
Perimeter = 4 * 4.24
Perimeter ≈ 16.96 feet (rounded to two decimal places)
Therefore, the perimeter of the inscribed square, with a radius of 3 feet, is approximately 16.96 feet.