An arc is a part of the circumference of a circle. We can obtain arcs in two different ways. One way is when there is a chord in a circle (see below).

On the diagram above, we have two arcs. The longer arc is known as the major arc (this is the orange arc) and the shorter arc is known as the minor arc (this is the green arc). For circles, major means larger and minor means shorter; we saw that this was the case when we were looking at segments. The larger segment was known as the major segment and the smaller segment was known as the minor segment).

he other ways that we can obtain an arc is when we have a sector. The arc is the curved part of the sector. On the diagram below, the arc is the orange line.

he other ways that we can obtain an arc is when we have a sector. The arc is the curved part of the sector. On the diagram below, the arc is the orange line.

**Working Out the Arc Length**

In an earlier section, we were looking at working out the circumference of a circle (click here to be taken through to that section). There were two different formulas for working out the circumference of a circle and the formulas are:

It does not matter what formula you use to work out the circumference of a circle. I think that the best one to use is the first formula and this is so that you do not get the formula for working out the circumference and the area of a circle confused. If you remember the first formula, then you can remember that working out the circumference of a circle involves the diameter (and not the radius). And then the formula for working out the area of a circle involves the radius (and not the diameter). However, it is totally up to you what formula you remember.

In order to work out the length of an arc, we need to multiply the circumference of the whole circle by the proportion of the circle that the sector is. For example, if we had half a circle, the arc length would be half of the circumference of the whole circle. Let’s have a few examples.

In order to work out the length of an arc, we need to multiply the circumference of the whole circle by the proportion of the circle that the sector is. For example, if we had half a circle, the arc length would be half of the circumference of the whole circle. Let’s have a few examples.

**Example 1**

What is the arc length in the sector below? Give your answer to 2 decimal places.

The angle of the sector is 135°. We are able to work out what proportion of the circle we have by dividing the angle of the sector by 360° (the angle for a whole circle).

Therefore, we have ^{3}/_{8} of the whole circle. The next step is to multiply the proportion of the circle that we have by the circumference of a whole circle.

*d*is the diameter of the circle. We are not given the diameter of the circle in the question, but we are able to work out the diameter by doubling the radius. The radius of the circle is 8 cm, which means that the diameter of the circle is 16 cm. We can now sub

*d*as 16 into the equation below and work out the arc length using a calculator.

The length of the arc is 18.85 cm.

**Generic Formula**

The generic formula for working out an arc length for a sector is given below.

**Example 2**

What is the arc length for the sector below? Give your answer to 2 decimal places.

We are going to be using the generic formula for working out the arc length. The generic formula is:

The angle of our sector is 290°. The equation above uses the diameter. We are given the radius, which means that we are able to obtain the diameter by doubling the radius. The radius is 4 cm, which means that the diameter is 8 cm. Let’s sub these values into the above formula.

The arc length is 20.25 cm.

**Working Backwards**

Sometimes we will be given the arc length and asked to work out the radius of the sector or the angle of the sector. We answer questions like this by setting the formula for working out the arc length equal to the arc length. We then solve to find either the radius of the sector or the angle of the sector.

Example 3

Example 3

The exact length of the arc below is 7.5π. What is the radius of the sector?

To answer this question, we are going to use the general formula for working out the arc length of a sector and work backwards to find out what the radius is; we will actually be using the formula to work out the diameter and then we will divide the diameter by 2 to obtain the radius. The general formula is:

Let’s sub the values that we are given in the question into the above formula. We are told that the arc length is 7.5π and the angle is 225°. We sub these two values into the formula.

When we look at the formulas, we notice that there is a π on either side of the equation and this means that we are able to divide both sides by π. The equation becomes:

The fraction can also be simplified.

We want to get rid of the fraction, and the best way to do this is to multiply both sides of the equation by the denominator of the fraction; we multiply both sides by 8.

The next step is to divide both sides by 5 because we want to find the value of

*d*and not 5*d*.The diameter of the sector is 12. However, we are trying to find the radius and the diameter is double the length of the radius. This means that we need to half the diameter to find out what the radius is.

The radius of this sector is 6 cm.