Arc Length

Objective

Now let’s get into Circles

So, now that we’ve gone over enough about triangles and quadrilaterals, let’s move on to circles.

So, circles are a little more complicated, and they have some specific properties.

But before we can actually go over problems dealing with circles, we need to go over some of their properties.

So, let’s start with some properties of circles:

Properties of Circles

So, there are a few things to mention when it comes to circles:

- The radius of a circle is the distance from the center of the circle, to the edge of the circle

- The diameter of a circle is the distance from one edge of the circle, to the other edge of the circle

- Because the length of circles are hard to measure, we measure how long a circle is from one point, along the circle, and back to the same point, by using the radius or the diameter, and we call this length the circumference.

So now that we have some of the properties of circles, we need to go over some important definitions.

So, here’s some more vocab:

Vocab:

So before we start off, we need to understand some vocabulary.

Some of it is a review, and some isn’t, so:

Radius – the radius of the circle is any distance from the center of the circle, to the edge of the circle.

Basically the radius starts at the center, and moves out to the edge, like:

Circumference – the circumference of a circle is the measurement of one full rotation around the circle.

Arc – an arc of a circle is an unbroken part of a circle consisting of two points called the endpoints, and all the point on the circle between them.

So, basically it’s a portion of the edge of the circle.

Like so:

So, by this logic, Arc Length is the length of an arc of a circle.

Finding the Circumference

Since we now know the arc length is the measure of a portion of the circle, we know that to find it, we need to know what the whole length of the circle is.

Then we can figure out what portion of the circle is the arc we are looking for, then we can find the arc length.

So, to start, let’s go over what the circumference of a circle is.

As stated before, the circumference of a circle is the length of the edge of the circle.

So, to explain this, let’s start with a string of 4 cm:

And let’s say we take that 4cm string, and bend it into a circle:

4cm

We know if we cut the circle in half (through the middle) we’ll get the diameter.

And now, we’ll measure the diameter, to see what we get:

1.27 cm

So, we have our diameter and our circumference, but we want to find a way to get both without doing this each time.

Well, what if we divided them?

Then maybe, we could get a constant and just do simple math instead of measuring each time.

So:

Or, in other words:

So, putting it altogether, we get:

Solving for C we get:

And since D is actually 2 times the radius, we get:

GREAT! SO WHAT DOES THIS HAVE TO DO WITH ARC LENGTH

Well, the arc length is basically just a portion of the circumference.

This means that if we have the circumference, and we know what the portion of the arc is, we can find the arc length by multiplying it by that portion.

It seems way more complicated than it actually is, so let’s start off with an example:

EXAMPLE 1:

Find the Arc Length for the following:

3 cm

So this looks a little intimidating, however, let’s look at this:

We know that the radius of this circle is 3cm

Which means we can find the circumference.

So, let’s start with that:

But what we need is just that smaller portion.

So what we need to do is just isolate that one portion.

So how do we do it?

Well, we need to figure out what the proportion of the part is that we need to find.

To do that, we need to figure out what the proportion of the area they gave us is.

Then we need to take our proportion, and multiply by it by the circumference of the circle.

Like so:

Which is about: 2.36 cm

AND THAT’S HOW WE FIND THE ARC LENGTH

However, it is sort of weird that we came up with a fraction multiplied by pi isn’t it?

Well, actually, that’s not as uncommon as you may have thought.

In fact, we have a name for this kind of measurement.

RADIANS

So what is a radian?

Well, a radian is another way to measure a degree, but in a way that makes better sense when dealing with circles.

Mainly because since we discovered the circumference of a circle deals with pi, it makes better sense to use pi to measure different parts of a circle.

So how do we get radians?

Well, it’s based off of what we call the unit circle:

-1

1

1

-1

So, the unit circle has a radius of 1 unit

Which means if we try to find the circumference of this circle we can see:

Now, plugging in 1 for r, we get:

So, we can conclude that:

​

SO, NOW THAT WE KNOW THE EQUATION

We can convert any degree to any radian we need.

The only thing we need to do is set up the proportion.

So for example:

Convert the following angles to radians:

So to convert this, we need to set up the proportions and solve:

And that’s how we solve for radians.

So, now that we have the equation, let’s look back at arc length:

Example 2:

Find the Arc Length for the following:

3 cm

So this looks a little intimidating, however, let’s look at this:

We know that the radius of this circle is 3cm

Which means we can find the circumference.

So, let’s start with that:

But what we need is just that smaller portion.

So what we need to do is just isolate that one portion.

So how do we do it?

Well, we need to figure out what the proportion of the part is that we need to find.

To do that, we need to figure out what the proportion of the area they gave us is.

Then we need to take our proportion, and multiply by it by the circumference of the circle.

Like so:

Which is about: 2.09

Example 3:

Find the Arc Length for the following:

4.5 cm

So this looks a little intimidating, however, let’s look at this:

We know that the radius of this circle is 4.5 cm

Which means we can find the circumference.

So, let’s start with that:

But what we need is just that smaller portion.

So what we need to do is just isolate that one portion.

So how do we do it?

Well, we need to figure out what the proportion of the part is that we need to find.

To do that, we need to figure out what the proportion of the area they gave us is.

Then we need to take our proportion, and multiply by it by the circumference of the circle.

Like so:

Which is about: 9.82 cm

Example 4:

Find the Arc Length for the following:

12 cm

So this looks a little intimidating, however, let’s look at this:

We know that the radius of this circle is 3cm

Which means we can find the circumference.

So, let’s start with that:

But what we need is just that smaller portion.

So what we need to do is just isolate that one portion.

So how do we do it?

Well, we need to figure out what the proportion of the part is that we need to find.

To do that, we need to figure out what the proportion of the area they gave us is.

Then we need to take our proportion, and multiply by it by the circumference of the circle.

Like so:

Which is about: 37.7cm

Example 5:

Find the Arc Length for the following:

13 cm

So this looks a little intimidating, however, let’s look at this:

We know that the radius of this circle is 3cm

Which means we can find the circumference.

So, let’s start with that:

But what we need is just that smaller portion.

So what we need to do is just isolate that one portion.

So how do we do it?

Well, we need to figure out what the proportion of the part is that we need to find.

To do that, we need to figure out what the proportion of the area they gave us is.

Then we need to take our proportion, and multiply by it by the circumference of the circle.

Like so:

Which is about: 61.3