# ACT Math : How to find the length of an arc

## Example Questions

### Example Question #1 : How To Find The Length Of An Arc

In the circle above, the angle A in radians is

What is the length of arc A?

Explanation:

Circumference of a Circle =

Arc Length

### Example Question #1 : How To Find The Length Of An Arc

The figure above is a circle with center at  and a radius of . This figure is not drawn to scale.

What is the length of the arc  in the figure above?

Explanation:

Recall that the length of an arc is merely a percentage of the circumference. The circumference is found by the equation:

For our data, this is:

Now the percentage for our arc is based on the angle  and the total degrees in a circle, namely, .

So, the length of the arc is:

### Example Question #521 : Plane Geometry

If a circle has a circumference of , what is the measure of the arc contained by a  degree angle located at the center of the circle?

Explanation:

A circle has a total of  degrees. If our angle is located at the center and is  degrees, we can do  to see that our angle makes up  of the complete circle.

Therefore, our arc is going to be  of our total circumference.

### Example Question #1 : How To Find The Length Of An Arc

What is the area of the sector of a circle with a central angle of  degrees and a radius of ? Simplify any fractions and leave your answer in terms of .

Explanation:

The formula for the area of a sector of a circle is:

The central angle given is 120 thus:

### Example Question #1 : How To Find The Length Of An Arc

A water wheel turns a  arc every minute. If the radius of the wheel is , how far in meters does the wheel turn along its edge each minute?

Explanation:

If the radius is , then the circumference of the wheel is:

If the wheel turns  each minute, then it turns  of the circumference each minute.

Thus, the wheel turns  each minute.

### Example Question #3 : How To Find The Length Of An Arc

The area of circle O is . The area of the shaded sector is . This figure is not drawn to scale.

What is the length of the arc ?

The total area of the circle is  and the area of the shaded region is .

Explanation:

If the area of the circle is , the radius can be found using the formula for the area of a circle:

For our data, this is:

Therefore,

Now, the circumference of the circle is defined as:

For our data, this is:

Now, we know that a sector is a percentage of the total area. This percentage is easily calculated:

So, the length of the arc will merely be the same percentage, but now applied to the circumference: