There are also two relationships between angles and circles which are crucial. First, let's define the term "central angle":
Central Angle: A central angle is an angle located at the center of a circle with endpoints on the circumference of that circle
Now, the crucial fact to learn is:
Central Angles and Arcs: In the following diagram:
The degree of angle AOB =
Thus, we can determine the degree of any central angle by looking at the length of its arc, and vice versa.
The circle below has a radius of 2 and has a length of . What is the degree of angle ?
The total circumference is so takes up of the total circumference, so the angle must be degrees.
The circle below has a radius of 6 and the degree of is 47 degrees. What is the length of ?
The total circumference is and the arc takes up of the total circumference. Thus, its length is .
A related fact to the above is:
Circumference Angle Theorem: In the following circle with radius :
The degree of angle ACB = .
This theorem is also called the Inscribed Angle Theorem.
In essence, if you have an angle on the circumference of a circle, its degree will be one-half the degree of a central angle with the same endpoints. In pictures:
Now, what may be surprising about this is that it does matter where on the circumference the angle begins from. In other words:
the angles and above have the same degree.
Finally, you must be careful to make sure that you are comparing the angle on the circumference to the correct central angle. For example:
and do not have the same degree. That is because they give rise to different central angles:
Thus, it is not enough to look at the endpoints of your angle. You also need to look at where on the circumference the angle is, so you can be sure to compare it to the relevant central angle.
1. The below circle has a center at . Find :
Since intersects a diameter of the circle, we know that the central angle there will be 180 degrees, and so by our Circumference Angle Theorem.
2. Find the value of
This is a bit of a trick question. Even though it looks like a right angle, there is no way to know that based solely on the above (see here). So the right answer would be that we cannot determine the value of
3. In the following diagram, Find
The total circumference is . Thus, our arc takes up of the total circumference. So
4. The radius of the following circle is Find
The total circumference of the circle is Thus, our arc takes up of the total circumference and thus
5. The radius of the following circle is 1. Find
Note that we have a right isosceles triangle, since the legs of the triangle are also radii of the circle and thus have a length of 1. So by the Reverse Pythagorean Theorem,
6. The radius of the below circle is 1. Find
Again, from the center, we would have a right isosceles triangle so the central angle of that arc must be 90 degrees. Thus,
7. In the below circle, which is larger?
C. They are equal.
D. It cannot be determined.
As noted above, it does not matter where on the circumference your angle is located; it just matters which central angle it corresponds to. In this case, both and have the same central angle. If the center of the circle is at point , then they both correspond to the central angle Thus, they are equal.