if the area of the square inscribed in a semicircle is 2cm^2,find the area of the square inscribed in a full circle
If the circle has radius r and the small square has side s, then
(s/2)^2 + s^2 = r^2
s^2 = 2
so,
1/2 + 2 = r^2
r = √5/2
Now, the larger square has diagonal 2r, so its area is 2r^2 = 2*5/4 = 5/2
wow - much smaller than I expected!
To find the area of the square inscribed in a full circle, we first need to determine the relationship between the area of the square inscribed in the semicircle and the area of the square inscribed in the full circle.
Let's assume that the side length of the square inscribed in the semicircle is 'a'. The area of this square can be calculated using the formula: Area = a^2.
Now, we know that the area of the square inscribed in the semicircle is 2 cm^2. Therefore, we have:
Area of the square inscribed in the semicircle = a^2 = 2 cm^2.
Next, we need to find the relationship between the side length of the square inscribed in the semicircle and the side length of the square inscribed in the full circle.
To do this, we can use the fact that the diameter of the full circle is equal to the diagonal of the square inscribed in the semicircle.
Let's assume that the side length of the square inscribed in the full circle is 'b'. The diagonal of this square can be calculated using the formula: Diagonal = b√2.
Now, let's consider the semicircle. The diameter of the semicircle is equal to the side length of the square inscribed in the semicircle plus the radius of the semicircle. The radius of the semicircle is equal to half the side length of the square inscribed in the semicircle, which is a/2.
Therefore, we have:
Diameter of the semicircle = a + a/2 = 3a/2.
Since the diameter of the semicircle is equal to the diagonal of the square inscribed in the full circle, we can write:
Diagonal of the square inscribed in the full circle = 3a/2.
Now, the diagonal of the square inscribed in the full circle is equal to b√2. Therefore, we can set up the following equation:
b√2 = 3a/2.
From this equation, b can be expressed in terms of a:
b = (3a/2) / √2.
Finally, we can calculate the area of the square inscribed in the full circle:
Area of the square inscribed in the full circle = b^2 = [(3a/2) / √2]^2.
Simplifying this equation will give us the final answer.