The area of a circle is found by: , where r is the radius.

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The area of the circle is .

The formula for the area, , of a circle with radius  is:

We can fill in 

You could do the arithmetic to get an area of about 1,017.876 square units, but it is ok and more precise to leave it as shown.

Half of the diameter 13 is the radius . Use the area formula:

Since it is stated that A is the center of the circle and ABCD is a square, angle BAD is a right angle that comprises exactly one fourth of the circle (90^o is one fourth of 360^o). This can also be seen through a proportion.

Since the angle that contains the red area is one fourth of the circle, finding the area of the entire circle, and then one fourth of that number gives the answer. Use the equation  to find the area of a circle, where  is area and  is the radius.

To find the area of the entire rink, find the area of the rectangle in the middle of the rink and add it to the area of the circle that would be formed by the two semi-crcles on the edges of that rectangle.

The area of the rectangle is found by multiplying its width by its length.

To find the area of the circle that accounts for the two semi-circles, use the formula , where  is the radius of the circle.

The radius of the circle is half of its diameter. The diameter of the circle is the same as the width of the rectangle in the case of this rink, and half of that is .

Adding this area to the area of the rectangle gives the area of the entire rink.

If the circumference is , then since  we know . We further know that , so 

The equation is already in a circle equation, and the right side of the equation stands for r^2 → r² = 81 and r = 9

The area of a circle is πr², so the area of this circle is 81π.

If you answered that the whirlpool’s area is 18.84 feet and therefore fits, you are incorrect because 18.84 is the circumference of the whirlpool, not the area.

If you answered that the area of the whirlpool is 56.52 feet, you multiplied the area of the whirlpool by 1.5 and assumed that that was the correct area, not the legal limit.

If you answered that the area of the backyard was smaller than the area of the whirlpool, you did not calculate area correctly.

And if you thought the area of the backyard was 30 feet, you found the perimeter of the backyard, not the area.

The correct answer is that the area of the whirlpool is 28.26 feet and, when multiplied by 1.5 = 42.39, which is smaller than the area of the backyard, which is 56 square feet. 

The total area of both circles is π + π = 2π

Since the region overlaps, we cannot count it twice, so we must subtract it.

we get 2π – π = π

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2.  Thus, the radius of the circle is half of the diameter, or 2√2.  The area of the circle is then π(2√2)², which equals 8π.  Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.