You know two things about the rectangle here: its area of 108 is equal to its length times its width, and the length and width are made of of diameters of the circles. The width is one diameter and the length is three diameters.  So you can express the area of the rectangle as:

108 = (Diameter)(3)(Diameter)

When you divide by 3, you see that the Diameter, squared, is equal to 36, meaning that the diameter is 6.

To convert from the diameter of a circle to its area, cut the diameter in half so that you have the radius. With a radius of 3, you can calculate the area:

In order to tackle this question, we’ll want to be able to effectively picture the situation at hand. If the pizza shop wants to know the smallest box that will exactly fit the pizza, we’re looking at a circle perfectly inscribed within a square, as shown below:

If the area of the pizza is  225 square centimeters, and area = r², then the radius of the circle must be 15, and the diameter must then be 30 centimeters. Since the diameter of the inscribed circle is equal to the length of each side of the square box, as shown below, the length of each side of the box must be 30 centimeters at a minimum to fit the pizza.

In order to answer this question, we’ll want to be able to visualize the situation at hand. If we want the largest circular rug that will fit exactly in a square room, we’re looking at a circle inscribed within a square, as shown below:

In order to find the area of the circular rug, we’ll need to recognize the connection between the circle and the square. Here, the diameter of the circle is equal to the length of each side of the square, as shown below:

So, if we can find the length of the side of the square, we’ve found our diameter, and can thus find the circle’s area! If the area of the square is 144 square feet and the area of a square = (side)², each side of the square must be 12 feet, making the diameter of the circle also 12 feet, and its radius 6 feet. From there, we can calculate the area as r², in this case, 6² or 36 feet, our correct answer.

An extremely important consideration on this problem is that the sizes are quoted in terms of the length of the diameter, but the area formula requires you to work with the radius.  So as you calculate the area of each pizza, it is very important to first divide the diameter by 2 so that you are working with the radius in the formula .

10-inch pizza → radius of 5 →  square inches

8-inch pizza → radius of 4 →  square inches

Therefore the difference in surface area is  square inches.

To calculate an arc length, you need two pieces of information:

1) The circumference of the circle (calculated as  or , where  is the radius and  is the diameter)

2) The measure of the central angle that connects the two end points of the arc.

Then the arc length is the proportion of the circumference represented by that angle: multiply  by the circumference and you have the arc length.

Here you're given the diameter (via line segment BD) as 18, so you know that the circumference is . And you know that angle AOB measures 40 degrees, so you can conclude that angle BOC is 140 degrees, since angles AOB and BOC must complete the straight line AC and straight lines measure 180 degrees.

Therefore your calculation is , which simplifies to .

The distance around the outside of a circle is, of course, the circumference. A common way to test the circumference in a word problem is to use the circumference of a wheel as a straight line distance: for each revolution of the wheel, a vehicle will travel the length of the circumference.

Here you know that the diameter of the wheel is 70 centimeters, which means that the circumference, calculated as , is . Note that the question asks for "approximately" the number of revolutions, and that the answer choices are spread quite far apart. This means that you can use an estimate of 3.14 or  for  and say that one revolution moves the wheel approximately 220 centimeters.  Since the wheel needs to cover 100000 centimeters, you should then divide 100000 centimeters by 220 to see that the answer is approximately 450.

This problem tests several of the core properties of circles:

Area = 

Circumference = 

Arc length = 

Here you're given the area, but to determine the perimeter of that sector you need to find the radius (for line segments AB and AD) and the arc length BCD. With an area of  that means that the radius is 12. Since you need two radii (AB and AD) to form the "legs" of the sector, that means that the straight-line legs sum to 24 (a good hint that your correct answer should include the number 24).

For the arc length, note that the central angle measures 45, and that . So the arc BD will equal one-eighth of the circumference. The circumference is , so you can find the arc by calculating  . Adding together the two legs plus the arc, you get your answer 

To calculate an arc length, such as the length of minor arc BC here, your job is to find the proportion that that arc represents out of the total circumference. So you'll need to find 1) the circumference and 2) the measure of the central angle of that arc.

Here since you know that the area is , you can use the formula  to determine that the radius measures 9.

Then you can plug in that radius into the circumference formula, , to find that the circumference measures 

From this point, you need to find the measure of angle BOC. Since angle AOC measures 180 degrees (you know that it's a straight line, because it's defined as a diameter), and angle AOB = 110, that means that BOC measures 70 degrees. So minor arc BC will be  of the total circumference, setting up the calculation:

That reduces to  which equals .

The cross section can be found by calculating the area of the larger, diameter 24, circle (inclusive of the pipe in gray) and subtracting the area of the smaller, diameter 22, circle (everything in white inside of the pipe). To do this, use the area formula: .

The radius of the larger circle is 12 and of the smaller circle is 11, meaning that your areas can be calculated as:

Outer/Larger Circle: 

Inner/Smaller Circle: 

Then subtract the areas and you'll be left with just the area of the gray ring: 

While this problem asks about probability, it's essentially an area of a circle problem. The probability of hitting the bullseye is just the area of the bullseye divided by the area of the dartboard in total.

Since  your job becomes to find the radius of each circle. You're quoted the diameters, so to find the radius just divide the diameter by 2.

Bullseye: Diameter = 4 so Radius = 2. Area = 

Dartboard: Diameter = 20 so Radius = 10. Area = 

So the probability of hitting the bullseye is  which reduces to .