How To Solve A Hard Geometry Problem That Stumps Many | Find x?#geometrypuzzle #geometry

追加: 2026-05-05

[00:00:00]Welcome amazing one and let's solve this beautiful geometric puzzle together. We are given a square that measures 10 units and inscribed are two equal semicircles such that the diameter of this lies on this part of this square and the diameter of this semicircle lies on this part of the square and you observe that they both over overlapped. The question is asking us find X. X is this part that is shaded. Now what does it mean? You see that this means that we are asked to find the area of this shaded region because area is the amount of space occupied by a PL shape and the amount of space that this X occupies is the same as the area. Now how do we get the area of the shaded region? It's just so easy.

[00:00:54]I'll take you through it step by step.

[00:00:56]Now because of this overlap you need to be careful in answering this. Let me show you how to get this area. So what you need to do is since these two semicircles are equal and the diameter is the same as the side of the square as well as this it means they both intersect. This point of intersection is the same as the center of the square.

[00:01:20]And because this is the center of the square it has made the work much easier for us. So it means that if we are to find the area of the shed, we cannot go straight to finding the this area of the semicircles and then subtract from the area of the entire square because there is an overlap. So what you need to do is let me show you what you would do.

[00:01:44]So what you need to do is we are free to draw a line from here from this center of the square to touch this just to get what we want. So if you do that, this is what you have. So you can join it this way.

[00:02:03]This is what you have. You can also do same here. So if you do it, this is what you have.

[00:02:15]Now what do you do with this? Observe closely. You will see that if I to find the area of this shaded it means that we need to find the area of this part which is let's label it A.

[00:02:32]If you find the area of this part A then find the area of this part the whole of this part which we can label part B then find the area of this part which is C.

[00:02:45]Then we subtract this three from the area of the entire square. And that gives us area of the square, the big square minus. So we're going to have area of the big square minus the area of A minus that of B minus the area of C. This is all we should do to find the area of the shaded. Now let's go back. You see that this is a square and for every square the sides are always equal. It means we're going to take it one at a time. So let's first get the area of the square.

[00:03:27]So if you do that we are going to have area of square is equal to if here is 10 this length is 10 here is 10 here is 10.

[00:03:37]So it means we are going to multiply two sides and that will give us 10 * 10. So it means area of the square is equal to 100 square unit. So what should we do next? Now that we have found the area of the square, we are going to find the area of this part A. Now how do you do that? If you look at this, you remember we said this is the center of the square and it has automatically divided this semicircle into two equal parts. In that case this has become the center of the semicircle. And when you divide a semicircle into two equal parts, it gives you a quarter circle. It means that this part of the shape A is the quarter circle. So how do you get the area of this quarter circle for the quarter circle? So we are going for area of shape A.

[00:04:31]So let's get the area of shape A. So to do that, let's take this part off. Area will now be area of your quarter circle which is 14 pi r² and remember that your r is the radius which is the line from the center to the circumerence. So a line from here to here is your radius.

[00:04:53]Now how do you get that? Remember that this total length is given as 10 units and this line has divided it equally. So it means from here to here is going to give us five units. In that case we are going to have the area will be 14 *<unk> * our radius is 5. So we have 5^ 2. And when you work this out it gives you area will be 5^ 2 is 25. So we have 25<unk> / 4. So this has given us the area of this quarter circle. So we can have it that this is the same as 25<unk>i / 4. Now what should we do next? We need to find the area of shape B. How do you do that? If you look at that shape E, you'll see that the major shape you are seeing is a square because from here to here is the radius which is the same as this and is five. From here to here you know is the same as from here down to this which is the same as five. So here is also five. from here to here is five.

[00:06:00]It means that from here down to this end is also five. So you see that this is a square. Now how do you get the area of that shape B? Let me show you. And if you are returning viewer, thank you for always choosing to learn to us. If you love what you do, why not subscribe for the more you get. Make sure notification bell is on because you'll get updates from us every day. Don't forget to like and share. Thank you. Let's keep doing this. So to find area of shape B we have it that area of shape B is the same as area of square which will give us just multiply two sides which is 5 squ and that will give us area is 25. So the area of shape B is 25. So this is 25 squar units. Now to find the area of shape C you see that shape C this is right angle there. Now sh this shape C is also a quarter circle and from here to here is five. So it means area of shape C area of shape C is the same as 14 *<unk> * the radius is 5^ 2 and that gives us 14 * 25<unk> which is the same as 25<unk> / 4. So this gives us 25<unk>i / 4. Now that we have found those areas, we can now find the area of the shaded. So it means that the area of shaded is the same as area of the big square which we said is 100 minus the area of shape A which is 25<unk> / 4 minus area of shape B which is 25 minus area of shape C which is 25<unk> / 4. Subtract this it gives us 75.

[00:07:56]So for this two is going to give us50.

[00:08:01]Add this it gives you 50<unk>i / 4.

[00:08:05]Let's simplify to have. So simplify this. We have two here is 2. 2 here is 25. We now have area of the shaded which is the same as x is equal to 75 - 25<unk>i / 2. So this is what we have for the area. You can also get the approximated value to have. So we have area of shed which is the same as x is approximately 35.73 and is in squared unit.

[00:08:51]This is also in squared unit. So these two areas are correct and I hope you learned something. Don't forget to tell us how much. Like share I also be curious to see your own method. Thank you for always choosing to learn yours.

[00:09:05]I look forward to seeing you in my next video. Have a lovely day. Bye.