A Nice geometry math question| can you find the area of the shaded portion

It's a freehand sketch, and I really struggled to get this drawing to this point, okay? Excuse me, so in this question, we are supposed to have a rectangle, okay? So this is a rectangle. rectangle, and this is a quarter circle, which is a quadrant, and this is a semicircle, so these are inscribed inside this rectangle, and we are asked to find this shaded portion. Now, this is Not easy, yes, so it's not easy, but it is easy. So let's do it together. First of all, we need to Analyze the problem and figure out where to start. What do I mean? What do we actually want to do? Now I want to find the area of the shaded portion. Let's find the area of the shaded portion, okay? Now, the area of the shaded portion is equal to the area of the rectangle, right? The area of this rectangle. Minus now, if we call A the area of the semicircle and we call B the area of the quarter circle okay, so if we subtract the sum of this from the area of the rectangle, we are going to get the area of the shaded portion. All right, so we are going to start by analyzing the area of the How do we get the area of a rectangle? The area of a rectangle is equal to length multiplied by breadth. the length is given as eight and this height is not given, not given, right? so that means that we cannot actually obtain the area of the rectangle at this moment, so let's work with what we have if we Call this point a, this point b, this point c, and this point d. Now, um, let us call the radius. Let's call the radius of the semicircle small r. So, in that case, let's call this part O. Okay, this meeting point: so AO is small r and DO is small r as well. Now remember that this is a rectangle, so as a result DA is equal to CB. Okay, so DA is equal to CB, which is equal to r plus r, which is 2r. All right, now if we call this place 2r, so it means that CB, which is the radius The radius of the quadrant is 2r, so the radius of the quadrant is 2r. Now remember a theorem that states that when two circles touch each other externally or internally, then if you join A straight line is formed connecting their centers with the meeting point, okay, a straight line. is formed when you connect the centers and their intersection points, so we are going to apply that Here, please click the subscription button and turn on the notification bell if this is your first time. The next time you see us, okay, give us a thumbs up if you're actually enjoying this video. And please turn at your convenience; there must be something new for you to watch, by God's grace. Okay, thank you very much. Don't just watch and move on—please leave a comment and share. I would truly appreciate it. That's okay, thank you very much. All right, now as I was saying, we are going to join the centers. Alright, okay, so um, it's going to be a freehand sketch, so don't expect anything perfect, so Alright, so this is our meeting point. Remember, this is K1, so this is the meeting point. of these two circles, and this is the center of the other circle. All right, now let us consider triangle OAB, so we have O, A, B; this is O, this is A, and this is B. Now remember that a o to this point is equal to the radius of this small circle and the radius of Sorry, the semicircle—now from here to here—happens to be the radius of this quadrant. but 2r equals the radius of the quadrant, so this distance is 2r from here to here, so two small Sorry, so it means that OB is r plus 2r. This is a right angle, right, since this is a rectangle. So point A is a right angle, and AB is 8. This is even now using the Pythagorean theorem. Using the Pythagorean theorem, we can see that r plus 2r squared is equal to OA. OA is our small one. so it's going to be r squared plus a b squared, which is x squared. Now r plus 2r is 3r, so we have 3r squared equals r squared plus 8 squared. Now remember that a multiplied by b raised to the fourth is equal to a squared multiplied by b squared, therefore 3r squared 3 squared times r squared equals r squared plus 8 squared. Now, 3 squared is 9. r squared equals r squared plus 64; now let's subtract r squared from both sides minus r squared minus r squared, so we have what this one will give us: r squared is equal to This is 0.64. All right, now let's divide both sides by 8. Divide both sides by 8, so we have 8r squared divided by 8 equals 64 divided by 8; now, divide 8 by 8 and we get r squared. is equal to 8, okay. Now this is very important: r squared is equal to 8. So let's write it somewhere. r squared equals 8; let's call it equation one. Now, if we take the square root of both sides take the square root of both sides, okay? So if we take the square root of both sides, we have The square root of r squared is equal to the square root of eight. Now, the square root of a all squared. is equal to a, okay, so that means that we can write this as r is equal to the square root of Eight is the square root of four multiplied by two, so r is the square root of four multiplied by the square. The square root of two, so r is two times the square root of two. Now let's write this somewhere: r is two times the square root of two. This is the equation. two now at this point we have gotten what we want, like very, very close to our solution, you're going to see that, all right now, if our r is two root two, it simply means that the height of a rectangle is two times two root two, okay? So you can say CB is equal to two r, so CB is equal to two multiplied by two root two, so CB is equal to four root two. Okay, all right. So that is what we have and we I'm going to write it somewhere; um, I'll just write it somewhere. CB is four root, so I know. You can't see this, I'm sorry. Now let's go back and then find the value of our area, okay? remember that the area of the shaded portion is the area of the rectangle, which we can find because We have the value of our arrow right now, but we cannot find a1 plus a2, so let's just try to find it. so a1 plus a2 equals now remember a1 is the area of the semicircle and the area of A semicircle is half pi r squared, okay, so we have half pi r squared plus now the area of the quadrant. is one over four pi r squared one over four pi now but the radius of the quadrant is two r Okay, two r squared, so we have a1 plus a2 equal to pi r squared divided by two. plus pi times two squared is four r squared divided by four, okay, so a1 plus a2 is equal to pi r squared over two plus now four, we divide it by itself to give us one. pi r squared will remain now the LCM of two and one is two a1 plus a2 is equal to two divides to one one times pi r squared is pi r squared plus one divides to two two times pi r squared is two pi r squared, so we have a1 plus a2 is equal to pi r squared plus two pi r. squared, we get three pi r squared, so we have three pi r squared divided by two. Now remember That our r squared is equal to 8, so write a1 plus a2 is equal to 3. a1 plus a2 equals three pi multiplied by our r squared, which is eight divided by two. Now, two we divide this to give us four, so we have a1 plus a2 is equal to three pi multiplied by four. So a1 plus a2 equals 12 pi. Yes, now we have arrived, we have reached our final destination. So the area of the shaded portion is equal to the area of the rectangle minus a1 plus a2. So the area of the shaded portion is equal to the area of the rectangle, which is length times height, right? or width that is eight times two r, and we got our r to be two root two, so it's going to be eight. multiplied by two multiplied by our r is two root two, okay, minus two root two, a1 plus a2 is 12 Pi. Okay, so we have the area of the shaded portion is equal to eight times two, which is 16, and 16 times two. Is 32 minus 12 pi okay in square units? All right, you can go ahead and press uh Include this in your calculator and get the actual value of a. Thank you so much for watching. See you in my next video.