A square of area 2 is inscribed in a circle. What is the area of the circle?
Well, isn't that a tricky little square! To figure out the area of the circle, we need to use some mathematical magic. Since the area of the square is 2, we know that each side of the square is √2.
Now, let's imagine that square behaving itself and staying inside the circle. The diameter of the circle would be equal to the diagonal of the square. Using a bit of Pythagorean theorem wizardry, we can find that the diagonal of the square is √2 times the square root of 2, which simplifies to 2.
So, the diameter of the circle is 2, which means the radius is 1. Now, the formula for the area of a circle is π multiplied by the radius squared. In this case, we have π times 1 squared, which means the area of the circle is... drumroll, please... π square units!
To find the area of the circle, we need to know the length of its radius. The square inscribed in the circle is formed by connecting the corners of the square. The diagonal of the square is equal to the diameter of the circle.
Since the area of the square is given as 2, we can find the length of its side by taking the square root of 2. Let's call this length "s".
s = √2
To find the length of the diagonal, we can use the Pythagorean theorem since the diagonal, the side of the square, and the other side of the square form a right triangle. The formula for the Pythagorean theorem is:
a^2 + b^2 = c^2
In this case, a = b = s, and c is the diagonal.
s^2 + s^2 = c^2
2s^2 = c^2
Taking the square root of both sides gives us:
√(2s^2) = c
√(2 * √2^2) = c
√(2 * 2) = c
√4 = c
c = 2
So the length of the diagonal (and therefore the diameter of the circle) is 2.
The radius of the circle is half of the diameter, so the radius is 2/2 = 1.
The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
Plugging in the value of the radius, we have:
A = π * 1^2
A = π * 1
A = π
Therefore, the area of the circle is π.
To find the area of the circle, we need to determine the radius.
First, let's find the side length of the square. Since the area of the square is given as 2, we can use the formula for the area of a square:
Area of a square = side^2
Substituting the given area, we have:
2 = side^2
To find the side length, we can take the square root of both sides:
√2 = side
Next, we need to find the diameter of the circle, which is equal to the diagonal of the square.
Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle.
To find the diagonal of the square, we can use the Pythagorean theorem:
Diagonal^2 = side^2 + side^2
Diagonal^2 = 2(side^2)
Substituting the value of the side length we found earlier, we have:
Diagonal^2 = 2(√2)^2
Diagonal^2 = 2*2
Diagonal^2 = 4
Taking the square root of both sides:
Diagonal = √4
Diagonal = 2
Therefore, the diameter of the circle is 2.
Now, we can find the radius of the circle by dividing the diameter by 2:
Radius = Diameter / 2
Radius = 2 / 2
Radius = 1
Finally, we can calculate the area of the circle using the formula:
Area of a circle = π * radius^2
Substituting the value for the radius, we have:
Area of a circle = π * 1^2
Area of a circle = π * 1
Therefore, the area of the circle is π square units.
Area=S^2=2, S=sqrt2=length of each side
of square, d^2=S^2+S^2=2+2=4, d=2=dia.
of circle, Ac=3.14r^2=3.14(1^2)=3.14=
area of circle.