A square is circumscribed about a circle. What is the ratio of the area of the circle to the area of the square?
A. 1/4
B. 1/2
C. 2/π
D. π/4
Well, it sounds like we have a shape party going on here! Let's see if we can figure out this ratio.
When a square is circumscribed about a circle, the circle is snugly tucked inside the square, touching all four sides. It's like a circle cozying up in a square blanket.
Now, to find the ratio of the circle's area to the square's area, we need to compare the sizes of these two shapes.
The area of a circle is given by the formula A = πr², where r is the radius. And since the circle is snugly tucked inside the square, the diagonal of the square is twice the radius of the circle.
Now, let's say the side length of the square is s. That means the diagonal of the square is 2s.
But hang on a minute! The diagonal of a square divides it into two 45-degree-angled triangles. Each of these triangles has a hypotenuse equal to the side length of the square.
So, if the diagonal is 2s, that tells us that the side length of the square is s√2. Pretty rad, right?
Now back to the circle's radius. Since the diagonal of the square is twice the radius, we can say that r = s/√2.
Finally, let's compare the area of the circle to that of the square:
Area of the circle = πr² = π(s/√2)² = πs²/2
Area of the square = s²
Now, let's take the ratio of the two areas:
(πs²/2) / s² = π/2
So, the correct answer is D. π/4. This means that the area of the circle is only a fraction of the area of the square. It's like the circle is photobombing the square, stealing its spotlight, but only a little bit.
To find the ratio of the area of the circle to the area of the square, we can use the formula for the area of a circle and the area of a square.
The area of a circle is given by the formula: A = πr^2, where r is the radius of the circle.
The area of a square is given by: A = side^2, where side is the length of a side of the square.
Let's assume that the side length of the square is s.
Since the circle is circumscribed about the square, the diameter of the circle is equal to the side length of the square (d = s).
The radius of the circle is therefore half the side length of the square (r = s/2).
Now we can calculate the area of the circle and the area of the square:
Area of circle = π * (s/2)^2 = π * (s^2/4)
Area of square = s^2
The ratio of the area of the circle to the area of the square is:
(π * (s^2/4)) / s^2 = π/4
Therefore, the correct answer is D. π/4.
To find the ratio of the area of the circle to the area of the square, let's start by figuring out the relationship between the radius of the circle and the side length of the square.
In a square, the diagonal is equal to the side length multiplied by the square root of 2. Since the side length of the square is equal to the diameter of the circle (because the circle is circumscribed about the square), the diagonal of the square is also equal to twice the radius of the circle.
Let's say the radius of the circle is r. In this case, the diagonal of the square is 2r. We can use the Pythagorean theorem to find the relationship between the diagonal and the side length of the square.
According to the Pythagorean theorem, the sum of the squares of the two shorter sides of a right triangle is equal to the square of the hypotenuse. In our case, the two shorter sides are the radius of the circle (r) and half of the side length of the square (s/2), and the hypotenuse is the diagonal of the square (2r).
So, we have:
(r^2) + (s^2/4) = (2r)^2
r^2 + s^2/4 = 4r^2
s^2/4 = 3r^2
s^2 = 12r^2
Now let's find the ratio of the areas:
The area of the circle is given by A = πr^2.
The area of the square is given by A = s^2.
Substituting s^2 = 12r^2, we get A(square) = 12r^2.
So the ratio of the area of the circle to the area of the square is:
A(circle) / A(square) = (πr^2) / (12r^2) = π/12.
Therefore, the correct answer is not listed among the options given.
Area of Circle = πr^2
Area of Square = (2r)^2 = 4r^2