# High School Math : How to find the length of an arc

## Example Questions

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

If a quarter of the area of a circle is , then what is a quarter of the circumference of the circle?

Explanation:

If a quarter of the area of a circle is , then the area of the whole circle is .  This means that the radius of the circle is 6.  The diameter is 12.  Thus, the circumference of the circle is .  One fourth of the circumference is .

### Example Question #21 : Circles

Figure not drawn to scale.

In the figure above, circle C has a radius of 18, and the measure of angle ACB is equal to 100°. What is the perimeter of the red shaded region?

18 + 10π

36 + 20π

36 + 36π

18 + 36π

36 + 10π

36 + 10π

Explanation:

The perimeter of any region is the total distance around its boundaries. The perimeter of the shaded region consists of the two straight line segments, AC and BC, as well as the arc AB. In order to find the perimeter of the whole region, we must add the lengths of AC, BC, and the arc AB.

The lengths of AC and BC are both going to be equal to the length of the radius, which is 18. Thus, the perimeter of AC and BC together is 36.

Lastly, we must find the length of arc AB and add it to 36 to get the whole perimeter of the region.

Angle ACB is a central angle, and it intercepts arc AB. The length of AB is going to equal a certain portion of the circumference. This portion will be equal to the ratio of the measure of angle ACB to the measure of the total degrees in the circle. There are 360 degrees in any circle. The ratio of the angle ACB to 360 degrees will be 100/360 = 5/18. Thus, the length of the arc AB will be 5/18 of the circumference of the circle, which equals 2πr, according to the formula for circumference.

length of arc AB = (5/18)(2πr) = (5/18)(2π(18)) = 10π.

Thus, the length of arc AB is 10π.

The total length of the perimeter is thus 36 + 10π.

The answer is 36 + 10π.

### Example Question #2 : How To Find The Length Of An Arc

Find the arc length of a sector that has an angle of 120 degrees and radius of 3.

Explanation:

The equation for the arc length of a sector is .

Substitute the given radius for  and the given angle for  to get the following equation:

Simplify:

Explanation:

### Example Question #4 : How To Find The Length Of An Arc

Find the circumference of the following sector:

Explanation:

The formula for the circumference of a sector is

,

where  is the radius of the sector and  is the fraction of the sector.

Plugging in our values, we get:

### Example Question #513 : Geometry

In the circle above, the angle A in radians is

What is the length of arc A?