Can you find area of the Sector? | (Find angle X) | |

[00:00:00]Welcome to Pre-Math. In this video, we have got this sector ACB with the center C, as you can see in this given diagram, such that this radius BC is three units, and likewise this radius AC is three units as well.

[00:00:21]And moreover, this arc AB length is four units, and furthermore, this angle ACB is being represented by X degrees. And now our task is to calculate the area of this sector. And furthermore, we are going to calculate the value of this angle X as well. Please don't forget to give a thumbs up and subscribe. And please keep in mind that this figure may not be 100% true to the scale.

[00:00:55]Let's go ahead and get started. And in this video, I'm going to share with you with two different approaches. So, therefore, please watch the video till the very end.

[00:01:03]And here's our very first method. And we are going to calculate this angle X first. And once we figured out the angle X, then we are going to calculate the area of this given sector. And now let's make an observation. We can see that this sector ACB is the part of the bigger circle. So, therefore, we are going to compare this sector with the full circle. In other words, we are going to use this proportion.

[00:01:42]The angle ratio is going to be equal to the arc ratio. And now let's recall the circumference of the circle. It is equal to two times pi times lower case R. And in our case, the radius lower case r is uh three units. So, therefore, the circumference of this circle is going to be 2 pi times our radius is three. So, that is going to be equal to 6 * pi. So, therefore, the circumference of the whole circle turns out to be 6 * pi. And likewise, the angle of the whole circle is 360°.

[00:02:38]And now we are going to fill in the blanks in this given proportion. And for this given angle ratio is going to be our this uh sector angle x divided by the whole circle angle is 360° is going to be equal to the arc ratio means this given arc AB is four. So, I'm going to write down four divided by the circumference of the whole circle is 6 * uh pi.

[00:03:15]And now I'm going to multiply both sides by 360.

[00:03:21]And I'm going to multiply 360 on the right-hand side as well to isolate x.

[00:03:27]And here we can see this 360 and 360 is gone. So, therefore, x is going to be equal to when we simplify and multiply the right-hand side, that is going to give us 240 divided by pi.

[00:03:48]And we know our pi value is approximately equal to 3.14.

[00:03:56]So, therefore, our angle X value is going to be approximately equal to 76.4 degrees.

[00:04:08]So, thus our angle X value turns out to be approximately equal to 76.4 degrees. And now in this next step, we are going to calculate the area of this sector.

[00:04:24]And now let's recall the sector area formula. Area is always equal to our angle theta divided by 360 degrees times pi r squared. Where our case r radius in our case is three units. And our angle is approximately equal to 76.4 degrees. And our angle X is also equal to 240 divided by pi. So, therefore, for this angle theta, I will be using this value 240 divided by pi.

[00:05:01]And our lower case r value as three.

[00:05:05]Let's go ahead and fill in the blanks.

[00:05:06]So, we got our angle is 240 divided by pi. So, I'm going to write down 240 divided by pi divided by 360 degrees times and then pi and then our radius is three whole squared.

[00:05:33]And now let's simplify furthermore. This whole thing could be written as uh 240 divided by 360 times pi * and this is going to be written as 9 * pi and here we can see this pi and pi is gone and we can see this 240 / 360 reduces to 2/3.

[00:06:07]So therefore we can write equals to 2/3 * 9. So therefore this sector area is going to be equal to 6 square units.

[00:06:22]So does the sector area turns out to be 6 square units and the angle is approximately equal to 76.4 degrees by using the very first method.

[00:06:35]And now let me share with you the second method as well. And in this method we don't need to calculate this angle X first. So therefore we are going to disregard this angle X and here in our case our arc length is 4. So therefore I'm going to call this one arc length and our radius BC is 3 units. So I'm going to call this one our radius and we are going to use this very simple formula this time.

[00:07:08]And the formula is the area of this sector is going to be equal to arc length / 2 * the radius. And here our arc length is 4 units whereas our radius 3 units. So let's go ahead and fill in the blanks in this formula. So the area of this sector is going to be equal to our arc length is 4 / 2 * the radius is 3. And if we multiply and simplify that is going to give us six square units, the area of this sector.

[00:07:51]So, this the sector area turns out to be six square units as well by using this second method. And that's our final answer.

[00:08:01]Thanks for watching and please don't forget to subscribe to my channel for more exciting videos. Bye.