Area of Sector of a Circle
The sector of a circle is the region bounded or enclosed by the two radii and the arc that they intercept. The sector of a circle resembles a triangle where the radii act as the two congruent legs, and the third side is the arc. It’s not really a side because it is a curved line segment. From our previous lesson, we should recognize that this arc is, in fact, a portion of the circumference of the circle. If we think of the circle as the whole pizza, the sector of the circle looks like a slice of pizza.
In the illustration below, sector XOY is bounded by the two radii which are the line segments [latex]\overline {OX}[/latex] and [latex]\overline {OY}[/latex] with the intercepted arc [latex] \overset{\Large\frown}{XY}[/latex].
Formula of the Area of Sector of Circle
To calculate the area of a sector, we use the formula for the area of the circle as the basis. The only difference is that instead of figuring out the area of a complete circle, we are figuring out the area of a portion or part of that circle.
There are two (2) variations of the formula to find the area of a sector. It depends on what unit of angle measure the central angle is given. It could either be in degrees or radians.
The most straightforward formula is when the central angle is given in degrees.
For the other formula with radians as the input value, we will perform some work. Note that the ratio of the central angle with the circle’s complete revolution is [latex]\large{\theta \over {2\pi }}[/latex]. If we multiply this to the formula of the area of the circle, we get
[latex]\large{\theta \over {2\pi }} \times \pi {r^2}[/latex]
That means we can cancel out [latex]\pi[/latex] which leaves us
[latex]\large{\theta \over {2\cancel{\pi} }} \times \cancel{\pi} {r^2}[/latex]
[latex]\boxed{\large{\theta \over 2} \times {r^2}}[/latex]
Here’s the formula:
Area of Sector Side-by-Side for Degrees and Radians
For easy reference, I placed the two formulas side-by-side so you can easily see which formula to use. Use the one on the left if the angle is measured in degrees while the one on the right if the angle is in radians.
Examples of Finding the Area of Sector
Example 1: Find the exact area of sector [latex]BOM[/latex].
The first thing that should cross our minds is that the problem wants an exact answer. If that is the case, we need to write our final answer in terms of pi, [latex]\pi[/latex]. In addition, note that the angle measure is in given degrees which means we are going to use the formula wherein the input value is degrees. That is the formula on the left as shown above.
Let’s plug in the values. Here, the angle is [latex]\theta = {70^\circ }[/latex] while the radius is [latex]r = 12[/latex].
Therefore, the exact area of sector [latex]BOM[/latex] is [latex]{7 \over 2}\pi[/latex] square inches.
Example 2: Calculate the area of a sector whose central angle is [latex]2.34[/latex] radians in a circle with a radius of [latex]6[/latex] feet.
The angle of the sector is given as radians. That means we are going to use the second version of the formula where the input value for the angle is in radians.
It is given that the angle is [latex]2.34[/latex] radians and the radius is [latex]6[/latex] feet. We substitute these values into the formula and then simplify.
Therefore, the area of the sector is [latex]42.12[/latex] square feet.
Example 3: Find the area of the sector below with a given arc measure in degrees. Use [latex]\pi = 3.14[/latex].
This problem is a little bit different than the previous two examples. The reason is that we are not given the measure of the central angle, instead, we are provided with the arc measure.
Remember that an arc (of the circle) subtends a central angle with the same arc measure. This implies that the measure of the central angle which intercepts the said arc has the same measure of [latex]120[/latex] degrees.
Since we know the measure of the central angle in degrees, and the length of the radius, we can now calculate the area of the sector using the first version of the formula as shown below.
Here’s the calculation of the area of the sector.
Therefore, the area of the sector is [latex]6.54[/latex] square centimeters.
Example 4: Find the central angle in degrees of a sector whose radius is [latex]13[/latex] yards and with an area of [latex]56[/latex] square yards. Use [latex]\pi = 3.1416[/latex].
The problem wants us to find the central angle in degrees which means we must use the formula of sector wherein the input value is degrees, not radians.
These are the known values.
radius = [latex]13[/latex]
area of sector = [latex]56[/latex]
[latex]\pi = 3.1416[/latex]
Let’s substitute the known values into the formula and then solve for the angle, theta.
Therefore, the central angle of the sector is [latex]37.97[/latex] degrees.
Example 5: A sector has an area of [latex]37[/latex] square meters. What is the measure of its central angle in radians if the radius has a length of [latex]8[/latex] meters?
This problem is very similar to problem #4. The only difference is the formula that we are going to use. We will use the second formula where the input value for the angle is in radians.
This formula doesn’t contain the value pi, [latex]\pi[/latex]. I just want to point that out before you start looking for it.
These are the known values from the problem.
radius = [latex]8[/latex]
area of the sector = [latex]37[/latex]
Let’s plug the values into the formula and then solve for the angle, theta.
Therefore, the central angle of the sector is about [latex]1.16[/latex] radians.
Example 6: The area of a sector is [latex]12[/latex] square kilometers. What is the length of the radius if the central angle is [latex]34[/latex] degrees? Use [latex]\pi = 3.14[/latex].
We are told about the area of the sector and the central angle it creates. What we want in this problem is to calculate the length of the radius.
To solve for the radius [latex]r[/latex], we just need to substitute the known values into the formula and then calculate for [latex]r[/latex].
These are the known values.
area of sector = [latex]12[/latex]
central angle = [latex]34[/latex]
Since the angle given is in degrees, we will use the first formula. We substitute the known values that we gather and then simplify.
Therefore, the length of the radius is about [latex]6.36[/latex] kilometers.
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