A Nice geometry math question|area of a circle|

We have an interesting geometry question to solve. Welcome back to my channel. Today we are asked to find the area of this particular circle, okay, if the radius of this circle is 5.6 units and the radius of this circle is 7.1 units. Please, this diagram is not drawn to scale and the center of this particular circle is at the center of this rectangle, all right, so meaning that from here to here is equal to from here to here. All right, these are the only information given to us. Now let's dive in and get the area of this circle. First, the area of a circle, what is the formula? The formula for area of a circle is pi r squared. Now if this is the center of this circle, okay, let's call this the center of the circle, if its radius is x, so it means that the area is pi x squared. So the problem we have now is how do we find the value of x? That is the only problem we have. Okay, now let us start by working with the information given to us. Why is this particular center in the middle of this line? Okay, it simply means that remember that from here to here is what the radius of this circle which is 7.1 and from here to here is the radius as well, since we are told that this is at the middle, so this is 7.1 as well. So it simply means that the side of this rectangle is 7.1 plus 7.1 which will give us 14.2 units. So we have 14.2 units here. Alright, now let us go to this side. The radius of this circle is 5.6. So that means that if I draw a straight line from here to the center, this is also 5.6 because from here to here also is the radius of this circle. So if I decide to draw a straight line like this and another straight line like this, it means that from here to here is 5.6 as well. Now let us draw a straight line to this and then another straight line from here. Okay, that means that remember that from here to here is x as well which is the radius of the smaller circle. So it means that from here to here is also x. Okay, now why am I doing all these things? We need to get an equation that has x in it and that is what I'm trying to do. Now if this is the center of this circle and this is the center of this circle, if we join them together like this, then drop a perpendicular from here to this place to the center of the circle as well. They are going to have 90 degrees here, right? Yes, and then from here to here is the radius of this circle which is 5.6. From here to here is x already, from here to here is x. Okay, so let me draw that line. Okay, so this is right angle, right? So that means from here to here will be 5.6 plus what? x. So here is 5.6 plus x. Alright, now for this one, this particular line here is going to be equal to 14.2, 14.2 minus 5.6, 5.6 minus x. Okay, so we have 12 minus 6 will give us 6, 13 minus 5 will give us 8. So this will be 8.6. So it means that 8.6 minus x, sorry, so it means that from here to here is 8.6 minus x. So I'm going to write that here. We don't need this anymore, right? So that we don't get confused. This is 14.6, 14.2, sorry. So from here to here is 8.6 minus x. Now it's remaining this side because I want to use Pythagorean theorem to solve this. Now for this particular line, if you look at this one, you notice that from here to here is 5.6, right? But from here to here is what, this is what we are looking for. But if you take from here to here, this is the radius of this small circle which is x. Okay, so it means that to get the remaining side here is going to be 5.6 which is the whole of this minus x. So this place is 5.6 minus x. And I think I've gotten all I needed. So let's go ahead and solve this. Okay, please click the subscription button, turn on your notification bell. If this is the first time you are seeing this amazing channel, all right, give us a thumbs up if you're actually enjoying this video. And tune in at your convenience. There must be something for you to watch by the grace of God, okay? All right, now this is what we have. So using the Pythagorean theorem, using Pythagorean theorem, okay, we have that 5.6 plus x squared is equal to 5.6 minus x squared plus 8.6 minus x squared. Now I'm going to make use of my calculator in solving this to save us time. So let me go and get one. Okay, now let's solve 5.6. Okay, first recall that a plus b all squared is equal to a squared plus 2ab plus b squared. And a minus b squared is equal to a squared minus 2ab plus b squared. Okay, now I think we don't need that diagram anymore. So let's turn it and use it to solve what we want to solve. Now from what we have here, we have 5.6 plus x squared. So it's going to be 5.6 squared plus 2ab plus 2 multiplied by 5.6 multiplied by x, right? Then plus b squared plus b squared, that is plus x squared, is equal to 5.6 minus x squared. It's going to be everything here, but you are going to have minus in the middle. So it's going to be equal to 5.6 squared minus 2 multiplied by, it seems this is short. That side is short for what I want to do. See what I'm going to do. I'm going to delete these two and start solving from here. By the time we get to this point, we must have simplified it further. Okay, so permit me to clean this and let's solve part of it here. Okay, so I'm going to copy what I got from here. So it's going to be 5.6 squared plus 2 multiplied by 5.6 multiplied by x, then plus x squared is equal to, then this is going to be everything here, but here will be negative, right? So we have 5.6 squared minus 2 brackets, 5.6 x plus x squared, then plus, this is going to be 8.6 squared, 8.6 squared minus 2 times 8.6 times x, okay? So 2 multiplied by 8.6 times x, then minus x squared plus x squared. All right, now it's time to simplify. So I think at this point, we can do that here. It will enter. Okay, so now let me clean part of this. I don't know why I'm still leaving it like that. Okay, so we continue. Now we have 5.6 squared. Okay, 5.6 squared is 31.36. 31.36 plus, now 2 times 5.6, 11.2 plus x squared is equal to, so this is going to be 31.36 as well, minus, this is going to be 11.2 plus x squared, then minus, this is going to be 8.6 times 2, 8.6 times 2, will give us a 17.2. Then, sorry, we are going to start with 8.6 squared first, okay? So 8.6 squared, will give us 73.96. So we have 73.96. We are what? We are adding. Then, minus, we got 17.2 here, 17.2x plus x squared. Okay, at this point, let us continue. But before that, let's just cancel out anything we have here and we also have it here. Let's cancel everything out to reduce the whole stress. So we have 31.6 here and we have 31.36 here. We have another thing. We have x squared and we have x squared. Yes. So let's write what we have left here. We have 11.2x is equal to minus 11.2x plus x squared plus x squared plus 73.96 minus 17.2x, okay? So we have 11.2x is equal to minus this minus this. Let's see what it will give us. I'm using my calculator because I don't want to make mistakes, okay? Or waste us a lot of time. So 11.2 and 17.2, right? Both of them will give us minus 28.4x. Then we have plus x squared and we have plus 73.96. So let us transfer this to this side so that we have zero here since we are about to form a quadratic equation. I hope you are actually enjoying this video. If yes, please don't forget to give us a thumbs up. Thank you. So if we transfer this to this side, let's write x squared first. So we have x squared. Then we have this with minus 28.4x. This will come to become minus 11.2x. Then we have plus 73.96 is equal to zero. So we have x squared. Now let's solve this. 28.4 plus 11.2. So this will be minus 39.6x plus 73.96 is equal to zero. Now at this point, let's also solve for x using the quadratic equation formula. If we use the quadratic equation formula, it shows that I'm going to get two values of x and we cannot have two values as our x, okay? So stay tuned and see how we choose the right value for our x. All right. So using the quadratic equation formula, x is equal to minus b plus or minus square root of b squared minus 4ac. Everything divided by 2a. Now our a is coefficient of x b coefficient of x minus 39.6 and c 73.96. Now let us substitute these values and solve for the values of our x. So we have x is equal to minus minus 39.6 becomes 39.6 plus or minus square root of b squared minus 39.6 squared then minus 4 multiplied by 1 multiplied by c which is 73.96. Everything divided by 2 multiplied by 1. So our x is equal to 39.6 plus or minus square root of now let's solve 39.6 squared 1568.16 minus 4 times 73.96. We have 295.84. Everything divided by 2. We are almost there, okay? Just stay tuned and see how we choose the right x for our x, okay? So x is equal to now we have 39.6 plus or minus the square root of 1568.16 minus 295.84. So we have 1272.32 divided by 2. x is equal to 39.6 plus or minus plus or minus we have square root of 1272.32. We have 35.67 approximately. Everything divided by 2. So the first value of x will be 39.6 plus 35.67 which is 75.27. 75.27 divided by 2 will give us an approximate value of 37.6. That is the first value of x. Then the second value of x will be equal to 39.6 minus 35.67. So it's going to give us 3.93. 3.93 divided by 2 gives us 1.97. That is an approximate value of our x, okay? Alright, now why are we getting the value of x? Remember that we are asked to find the area of this circle. By now looking at this circle you will know that the radius of this circle cannot be up to this. So it simply means the radius is approximately 1.97. Now let us go ahead. If there is any mistake in the calculation here please do correct me in the comment section because I know having so many numbers there maybe I might have typed something wrong. So the area of the circle is pi r squared is equal to pi is 3.142 multiplied by 1.97 squared. So let's solve this and get the answer, okay? So we have 1.97 squared will give us 1.97 squared will give us 3.8809. 3.8809 multiplied by 3.142 will give us an approximate value of 12.2 square units, okay? So this is the area of the smaller circle. Thank you so much for watching. Bye.