Area Sector & Arc Length

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, \theta, and the area of the shaded region, S. The proportion is:
\frac{\theta}{360}=\frac{S}{A}

For example:
\frac{90}{360}=\frac{S}{A}
\frac{1}{4}=\frac{S}{A}
This means that this shaded area, S, is one-fourth of the area of the entire circle.

For example:
If we had an angle that was 40\textdegree and the radius is 6 inches, then how would we solve this? Well, if we plug in our angles measure into our proportion, we have:
\frac{40}{360}=\frac{S}{A}

In order to calculate S, we first need to know the total area. We need to use our area of a circle formula:
A=\pi{r^2}
A=\pi{6^2}
A=36\pi

So:
\frac{40}{360}=\frac{S}{36\pi}

We can first reduce this to:
\frac{1}{9}=\frac{S}{36\pi}

Now we can cross multiply:
36\pi=9S
Divide both sides by 9:
4\pi=S

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:
\frac{\theta}{360}=\frac{L}{C}

For example:
If we had an angle that was 40\textdegree and the radius is 6 inches, then how would we solve this? Well, if we plug in our angles measure into our proportion, we have:
\frac{40}{360}=\frac{L}{C}

In order to calculate L, we first need to know the circumference. We need to use our circumference formula:
C=2\pi{r}
C=2\pi{6}
C=12\pi

So:
\frac{40}{360}=\frac{L}{12\pi}

We can first reduce this to:
\frac{1}{9}=\frac{L}{12\pi}

Now we can cross multiply:
12\pi=9L

Divide both sides by 9:
\frac{4\pi}{3}=L