# HiSET: Math : Arc length and area of a sector

## Example Questions

### Example Question #1 : Arc Length And Area Of A Sector

The depicted circle has a radius of 3 cm. The arc length between the two points shown on the circle is  cm. Find the area of the enclosed sector (highlighted in green).

Explanation:

First, find the area and circumference of the circle using the radius and the following formulae for circles:

Substituting in 3 for  yields:

Next, find what fraction of the total circumference is between the two points on the circle (the arc length).

Finally, use this fraction to calculate the area of the enclosed sector. Note that this area is proportional to the above fraction. In other words:

So the Sector Area is one sixth of the total area.

Cross multiply:

Divide both sides by 6, then simplify to get the final answer:

### Example Question #2 : Arc Length And Area Of A Sector

is the center of the above circle, and . Evaluate the length of .

Explanation:

The radius of the circle is given to be . The total circumference  of the circle is  times this, or

.

The length  of  is equal to

Thus, it is first necessary to find the degree measure of . If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,

Letting , since the total arc measure of a circle is 360 degrees,

We are also given that

Making substitutions, and solving for :

Multiply both sides by 2:

Subtract 360 from both sides:

Divide both sides by :

,

the degree measure of .

Thus, the length  of  is

### Example Question #3 : Arc Length And Area Of A Sector

is the center of the above circle, and . Evaluate the area of the shaded sector.

The information given is insufficient to answer the question.

Explanation:

The radius of the circle is given to be . The total area  of the circle can be found using the area formula:

The area  of the sector is equal to

Thus, it is first necessary to find the degree measure of . If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,

Letting , since the total arc measure of a circle is 360 degrees,

We are also given that

Making substitutions, and solving for :

Multiply both sides by 2:

Subtract 360 from both sides:

Divide both sides by :

,

the degree measure of .

Thus, the area of the shaded sector is

### Example Question #4 : Arc Length And Area Of A Sector

is the center of the above circle, and . Evaluate the area of the shaded sector.

The information given is insufficient to answer the question.

Explanation:

The radius of the circle is given to be . The total area  of the circle can be found using the area formula:

The area  of the sector is equal to

Thus, it is first necessary to find the degree measure of . If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,

Letting , since the total arc measure of a circle is 360 degrees,

We are also given that

Making substitutions, and solving for :

Multiply both sides by 2:

Add 360 to both sides:

Divide both sides by 2:

,

the degree measure of .

Therefore, the area  of the shaded sector is

### Example Question #5 : Arc Length And Area Of A Sector

is the center of the above circle, and . Evaluate the length of .

Explanation:

The radius of the circle is given to be

The total circumference  of the circle is  times this, or

.

The length  of  is equal to

Thus, it is first necessary to find the degree measure of . If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,

Letting , since the total arc measure of a circle is 360 degrees,

We are also given that

Making substitutions, and solving for :

Multiply both sides by 2:

Add 360 to both sides:

Divide both sides by 2:

,

the degree measure of .

Thus, the length  of  is

### Example Question #6 : Arc Length And Area Of A Sector

Refer to the above figure. Give the ratio of the area of Sector 2 to that of Sector 1.

Explanation:

The ratio of the area of the larger Sector 2 to that of smaller Sector 1 is equal to the ratio of their respective arc measures - that is,

.

Therefore, it is sufficient to find these arc measures.

If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,

Letting , since the total arc measure of a circle is 360 degrees,

We are also given that

Making substitutions, and solving for :

Multiply both sides by 2:

Add 360 to both sides:

Divide both sides by 2:

,

the degree measure of .

It follows that

By the Arc Addition Principle,

Since , the central angle which intercepts , is a right angle, . By substitution,

,

and

The ratio  is equal to

,

a 5 to 1 ratio.