In this video, we are going to look at area sector and arc length.

Here we have two radii drawn from circle *O* that are perpendicular to each other, so that the angle formed is a right angle. That is one-fourth of the whole circle. If we wanted to find the shaded area, we would expect it to be one-fourth of the entire area of the whole circle. If we had a diameter drawn instead, then the new shaded area would represent half of the area of the circle. There is a direct relationship between the angle and the area of the circle. This relationship is a proportion that exists between the angle that we have, , and the area of the shaded region, . The proportion is:

**For example:**

This means that this shaded area, , is one-fourth of the area of the entire circle.

**For example:**

If we had an angle that was and the radius is inches, then how would we solve this? Well, if we plug in our angles measure into our proportion, we have:

In order to calculate S, we first need to know the total area. We need to use our area of a circle formula:

So:

We can first reduce this to:

Now we can cross multiply:

Divide both sides by 9:

We can also use the angle measure to help us find the arc length. Just like how the shaded area was a fraction of the total area, the arc is a fraction of the total circumference. This formula is:

**For example:**

If we had an angle that was and the radius is inches, then how would we solve this? Well, if we plug in our angles measure into our proportion, we have:

In order to calculate L, we first need to know the circumference. We need to use our circumference formula:

So:

We can first reduce this to:

Now we can cross multiply:

Divide both sides by 9: