To find arc length, multiply the total circumference of a circle by the the fraction of the total circle that defines the arc with the length for which you are solving.

In this case, to find the total circumference:

To find the fraction with which we are concerned, make a fraction with the number of degrees in  in the numerator and the total degrees in a circle in the denominator:

Multiply together and simplify:

Using the formula for arc length, we can plug in the given angle and radius to calculate the length of the arc that subtends the central angle of the sector. The angle, however, must be in radians, so we make sure to convert degrees accordingly by multiplying the given angle by  :

Examine the figure below, which shows the arc and chord in question.

If we extend the figure to depict the circle as the composite of four quarter-circles, each a  arc, we see that  is also the side of an inscribed square. A diagonal of this square, which measures  times this sidelength, or

,

is a diameter of this circle. The circumference is  times the diameter, or

.

Since a  arc is one fourth of a circle, the length of arc  is

, so, by the Multiplication Property of Inequality,

.

The degree measure of an arc is twice that of the inscribed angle that intercepts it, so the above can be rewritten as 

.

Here we need to recall the total degree measure of a circle. A circle always has exactly  degrees. 

Knowing this, we need to utilize two other clues to find the degree measure of .

1) Angle  measures  degrees, because it is made up of line segment , which is a straight line.

2) Angle  can be found by using the following equation. Because we are given the fractional value of its area, we can construct a ratio to solve for angle :

So, to find angle , we just need to subtract our other values from :

So, .

Call the length of a side of Square B . Its perimeter is , which is the radius of Circle A.

The area of the circle is ; that of the square is . Therefore, a sector of the circle with area  will be  of the circle, which is a sector of measure

This is the kind of question we can't get right if we don't know the trick. In a circle, the size of an angle at the center of the circle, formed by two segments intercepting an arc, is twice the size of the angle formed by two lines intercepting the same arc, provided one of these lines is the diameter of the circle. in other words,  is twice .

Thus,

We can see that the points devide the  of the circle in 5 equal portions.

The final answer is given simply by  which is , this is the angle of a slice of a pizza cut in 5 parts if you will!

As we have seen previously, the 6 points divide the  of the circle in 6 portion of same angle. Each portion form an angle of  or 60 degrees. As we also have previously seen, the angle formed by the lines intercepting an arc is twice more at the center of the circle than at the intersection of the lines intercepting the same arc with the circle, provided one of these lines is the diameter. In other words, . Since  is 60 degrees, than,  must be 30 degrees, this is our final answer.  

A square with area 100 would have a side length of 10, which is the diameter of the circle.  The area of a circle is , so the answer is .