The circumference of a circle is equal to the diameter of the circle times . The diameter is equal to twice the radius so:

The radius of the first can be solved as follows:

Likewise for the second circle:

The are of a circle is given by the following formula:

.

The ratio of the larger area to the smaller area can be found as follows:

Cancelling out  and dividing gives the correct answer of 

This question asks you to apply the concept of area in finding both the area of a circle and square. Since the cirlce is inscribed in the square, we know that its diameter (two times the radius) is the same length as one side of the square. Since we are given the radius, , we can find the area of both the circle and square.

Square:

This gives us the area for the entire square.

The bottom half of the square has area .

Now that we have this value, we must find the area that the circle occupies. The area of a circle is given by .

So the area of this circle will be .

The bottom half of the circle has half that area:

Now that we have both our values, we can subtract the bottom half of the circle from the bottom half of the square to give us the shaded region:

To solve, simply use the formula for the area of a circle. Thus,

To remember this formula, think about area and how it is in two dimensions, not one or three. So, one of our variables will have to be squared. By process of elimination, it must be r because pi doesn't change so it only makes sense to square the variable that changes between all circles.

The circumference C of a circle with radius r is expressed by:

Our circle's circumference is . Therefore,

Plug this r-value into the formula for the area of a circle.

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for  in the formula :

 - this is the area of the white circle in the center.

 - this is the total area of the figure.

The area of the white ring is the difference between those at of the second-largest and third-largest circles:

The total area of the white portion of the figure is 

, 

which is 

of the figure.

If the radius of the circle is  units, that means that the diameter is  units. Because the circle is tangent to the sides of the square, we also know that each side of the square is also  units. 

The area of the square is given by its length times its width, or:

The area of the circle is given by:

The area that we're looking for lies outside of the circle but inside of the square. To get this area, we subtract the area of the circle from the area of the square. This gives us our answer of  square units. 

If the area of the square is , then that means that each side is  since the area of a square is given by . 

In the figure, the side of the square is equal to the diameter of the semicircle. Therefore, the semicircle's radius is . 

The area of the semicircle is given by:

The area of the entire figure is the sum of the two component areas. 

Our total area is . 

Below is the figure referenced; note that the hexagon is divided by its diameters.

The hexagon is divided into six equilateral triangles; the length of each side is equal to the radius of the hexagon, and, consequently, the radius of the circle. The regular hexagon has perimeter 120, so the length of each congruent side is . The radius of the hexagon, and of the circle, is 20 also. 

The area of the circle can be found using the area formula, setting :

Below is the figure referenced; note that the hexagon is divided by its diameters, and that an apothem - a perpendicular segment from the center to one side - has been drawn.

The hexagon is divided into six equilateral triangles. The regular hexagon has perimeter 120, so the length of each congruent side is .

Specifically,  is a 30-60-90 triangle.  is the midpoint of , so 

By the 30-60-90 Theorem, since  and  are the short and long legs of a right triangle, 

.

This is the apothem of the hexagon; this is also the radius of the circle. 

The area of the circle can be found using the area formula, setting :