If area of a square inscribed in a semi circle is 2 square centimeter. Fine the are of the square inscribed in a full circle
To find the area of the square inscribed in a full circle, we can use the relationship between the areas of a square and a circle.
First, let's find the area of the square inscribed in a semi-circle, which is given as 2 square centimeters.
We know that a circle contains 360 degrees, and a semi-circle contains half of that, which is 180 degrees.
Now, let's consider the square inscribed in the semi-circle. The square is inscribed in a quarter of the circle because the diameter of the semi-circle passes through the center of the square, dividing the square into four equal sections.
Since a square has four 90-degree angles, one of the 90-degree angles will fall on the diameter of the semi-circle. Therefore, the quarter circle will have an arc of 90 degrees.
So, the area of the quarter circle can be found by multiplying the radius squared by π (pi), and dividing it by 4, because we only consider a quarter of the circle.
Let's assume the side length of the square inscribed in the semi-circle is "a". The radius of the circle is half the length of the side of the square, which is "a/2".
Therefore, the area of the quarter circle is:
(1/4) * π * (a/2)^2
We are given that this area is equal to 2 square centimeters:
(1/4) * π * (a/2)^2 = 2
To find the area of the square inscribed in the full circle, we need to find the side length, "a", and then square it.
Let's solve the equation for "a":
(1/4) * π * (a/2)^2 = 2
First, simplify the equation:
π * (a/2)^2 = 8
Now, multiply both sides by 4:
π * (a/2)^2 * 4 = 8 * 4
π * a^2/4 = 32
Multiply both sides by 4/π:
a^2 = 128/π
Finally, to find the area of the square inscribed in the full circle, square the side length "a":
Area = a^2 = (128/π)^2 square centimeters
So, the area of the square inscribed in the full circle is (128/π)^2 square centimeters, which can be approximated as 163.67 square centimeters.