To solve this problem, you must find the area of the entire circle (garden and sidewalk) and subtract it by the area of the inner garden. The entire area has a radius of 5 ft. (3 ft. radius of the garden plus the 2 ft. wide sidewalk), giving it an area of . The inner garden has a radius of 3 ft. and an area of . The difference is , which is the area of the sidewalk. 

To find the area of the garden, you need to find the entire area and subtract that by the area of the inner circle, or the monument. The radius of the larger circle is 50, which makes its area . The radius of the inner circle is 30, which makes its area . The difference is .

Since the answers are in terms of pi, simply find the area of each circle in terms of x and ∏:

Smaller: ∏(5)² = 25∏

Larger: ∏x²

We must subtract the inner circle from the outer circle; this translates to ∏x²-25∏.

There are two steps to this problem: determining the area of the circle and determining the area of the square. The area of the circle is πr² which is π(2/1)² or π. AD is a diameter of circle O and creates two isosceles right triangles with ACD and ABD. The relationship between sides of an isosceles right triangle is 1 : 1 : √2. Thus the sides of square ABCD are √2 and the area is 2. The area of the shaded region is the area of the circle minus the area of the square, or π – 2.

Therefore the 16 inch pizza is the better deal.

To find the radius of circle B, use the circumference formula (c = πd = 2πr):

2πr = 36π

Divide each side by 2π: r = 18

Now, if circle A has a radius half the length of that of B, A's radius is 18 / 2 = 9.

The area of a circle is πr².  Therefore, for A, it is π*9² = 81π.

We know the triangle inscribed within the circle must be isosceles, as it contains a 90-degree angle and fixed radii. As such, the opposite angles must be equal. Therefore we can use a simplified version of the Pythagorean Theorem, 

a² + a² = c² → 2r² = 16 → r² = 8; r = √8 < 3. (since we know √9 = 3, we know √8 must be less); therefore, Quantity B is greater. 

A radius of 5 means we need a distance of 5 from the center to any points on the circle. We need 5² = (1 – x₂)² + (2 – y₂)². Let's start with the first point, (3,4). (1 – 3)² + (2 – 4)² ≠ 25. Next let's try (4,6). (1 – 4)² + (2 – 6)² = 25, so (4,6) is our answer. The same can be done for the other three points to prove they are incorrect answers, but this is something to do ONLY if you have enough time. 

This question is easy on the whole, though you must not be intimidated by one small fact that we will soon see. Set up your standard circumference equation:

The circumference is  feet, so we can say:

Solving for , we get:

Some students may be intimidated by having  in the denominator; however, there is no need for such intimidation. This is simply the answer!

Since we know that the area of Square  is , we know , where  is the length of one of its sides. From this, we can solve for  by taking the square root of both sides. You will have to do this by estimating upward. Therefore, you know that  is . By careful guessing, you can quickly see that  is . From this, you know that the diameter of your circle must be half of , or  (because it is circumscribed). Therefore, you can draw: