# Precalculus : Find the Area of a Sector Using Radians

## Example Questions

### Example Question #1 : Find The Area Of A Sector Using Radians

Find the area of a sector if the radius of the circle is 4, and the angle of the sector is  radians.

Explanation:

Write the formula for the area of the sector in radians.

### Example Question #1 : Find The Area Of A Sector Using Radians

Find the area of a sector if the radius is 1 and the angle of sector is  radians.

Explanation:

Write the formula for the area of the sector in radians and substitute the given values.

### Example Question #3 : Find The Area Of A Sector Using Radians

If the diameter of a circle is 6, find the area of a sector with a sector angle of .

Explanation:

Write the formula for the area of the sector in radians.

The given diameter is 6, which means the radius is 3.

Substitute both the radius and theta to solve for the area.

### Example Question #1 : Find The Area Of A Sector Using Radians

Find the area of a sector with the radius of 1 and angle of .

Explanation:

Write the formula for the area of a sector in radians.

Substitute the radius and the angle.

### Example Question #1 : Find The Area Of A Sector Using Radians

Find the area of a sector with a radius and angle of .  Express the answer in terms of .

Explanation:

Write the formula for the area of a sector.  Substitute the radius and angle to solve for the area.

### Example Question #1 : Find The Area Of A Sector Using Radians

Find the area of a sector with a radius of 10 and an angle of  radians.

Explanation:

Write the area of the sector in radians.

Substitute the givens and solve.

### Example Question #7 : Find The Area Of A Sector Using Radians

Katelyn is making a semi-circular design to put on one of her quilts. The design is not a perfect half-circle however, she needs to make the central angle  radians. If the radius of the circle is , what is the area of the semi-circular design?

Explanation:

Katelyn is making a semi-circular design to put on one of her quilts. The design is not a perfect half-circle however, she needs to make the central angle  radians. If the radius of the circle is , what is the area of the semi-circular design?

Recall the following formual for area of a sector:

So, we plug in our knowns and solve for area!

### Example Question #8 : Find The Area Of A Sector Using Radians

Lucy is making a solar panel to cover a portion of a satellite dish. If the central angle of the sector the solar panel will cover is , and the satellite dish has a radius of , what area will the solar panel cover?

Explanation:

Lucy is making a solar panel to cover a portion of a satellite dish. If the central angle of the sector the solar panel will cover is , and the satellite dish has a radius of , what area will the solar panel cover?

To calculate area of a sector, use the following formula:

Where the numerator of the fraction is the measure of the desired angle in radians, and r is the radius of the circle.

Now, we know both our variables, so we simply need to plug them in and simplify.

Now, this looks messy, but we can simplify it to get:

Next, use your calculator to find a decimal answer, and then round to get our final answer.

Making the area of the sector

### Example Question #9 : Find The Area Of A Sector Using Radians

Find the area of a sector in a circle, given that it encompasses  of the actual circle, with a circle diameter of

Explanation:

Equation for sector area  is given by

, where  is the angle measure of the sector in radians, and  is the radius of the circle.

In our case, the sector encompasses  of the circle or

To determine the radius , given diameter

The sector area therefore is: