There's an even simpler solution: 1) There are Yellow, Green and Pink rectangles under the Orange one. That means, that the combined rectangle YGP (containing the Yellow, Green and Pink rectangles) have the same width as the Orange one. 2) The area of the YGP rectangle is 3 times greater than the area of the Orange rectangle (since it's a composite of 3 rectangles of same area) 3) Since Orange and YGP have the same width and the are ratio of 1/3, that means that their height ratio is also 1/3, resulting in the YGP height is 9 4) The side of the Square is 3 + 9 = 12, and the area is 144
When you have fancy tools, everything looks like a fancy job.. But when you want to get the job done, the simplest solution is best. This is the best solution.
@@therealdoc All of the squares have the same area. If you get rid of the blue section you have the combined area of the Yellow Green and Pink sections is 3 times that of the orange section with the same width, therefore they must have height 3 times greater than orange to have an area 3 times greater. This means their height must be 9 making the combined height of the larger square 9+3=12 and then 12*12 is 144.
My solution was Yellow+Green+Pink = 3x Orange, so I must be able to place 3 other Oranges bellow the one given, so I am left with 4x orange and a blue. each orange has the side of 3, so 4x3 = 12 = x ...
i did it an easier way. The yellow, green, and pink rectangles make a big rectangle with unknown lengths. Lets call those lengths x and y. The area of that rectangle is then xy. Area of orange rectangle = 3x. we know that the area of the orange rectange is 1/3 times the area of the combined rectangle. therefore 3(3x) = xy. 9x = xy. y = 9. Therefore the length of one side of the square is 3 + 9 = 12. 12 * 12 = 144.
I like your videos - straight forward, not wasting time rewriting equations ad infinitum to add minutes. I also solved this the simpler way it's been discussed in comments, but also have a small problem with the presentation. Put simply, we had to assume the smaller portions were all rectangles, even though it wasn't stated.
I love problems where there's many different ways to solve. Here's the fastest I found: Let "A" be the area of a colored rectangle. The bottom side of the orange rectangle must be A/3. Then, let "x" be the height of the yellow rectangle. The bottom of the orange rectangle and left side of the yellow rectangle can be the perpendicular sides of a new rectangle, with area x*A/3. This rectangle is the same as the yellow, green, and pink rectangles combined, so we know it also has an area of 3A. Ax/3=3A, therefore x (the height of the yellow rectangle) is 9. Simple arithmetic from there, 9+3=12 and 12²=144.
Simpler Solution: Area of orange box = 3x, where x is the unknown side of the orange box Area of all boxes but the blue = x * the side length of the whole square So the side length of the whole square = the area of all but blue divided by the top side of the orange box, x: 12x / x = 12 Square this to get the area of the whole square: 12^2 = 144
Another solution Green and pink are equal, because areas and horizontal sides also are. So as yellow vertical is 2*(green or pink) vertical, and the areas are the same, yellow horizontal its ½ green vertical
So, orange vertical is 3* yellow horizontall. Then, as areas are the same, 3*orangeV = 3*3*yellowH = yellowH*yellowV, so yellowV is 9. So then, the side of the square is 3+9=12 length
Since there are three rectangles under the green one, the length of the yellow one must be 3 * 3 = 9. So the side of the square must be 3 + 9 = 12, and, therefore, its area 12² = 144 square units.
I solved it a different, and I think easier way: Ignore the blue rectangle. If you re-shape the orange, green and pink rectangles so they're the same shape as the orange one while preserving their area, all four will be exactly the same as the orange one... therefore there's 4 that can fit into the whole square minus the blue rectangle, therefore the side length is 4 x 3 = 12
Another algebra-free way. 1. Green and pink are the same width so, because they're the same area, they're also the same height as each other, ie they're identical. 2. Yellow is twice as high as either green or pink so is half as wide as them. 3. Orange is three times the width of yellow because it's as wide as yellow and green, and green is twice as wide as yellow. 4. If orange is three times as wide as yellow, it's a third of the height, so yellow has height 9, and the square height 12. 5. You can use the fact we know that it's square to get the answer, 144.
I got there through another way. First, I let the width of the orange rectangle be x, and the side of the big square be y. Then, since the rectangle formed by yellow, green and pink shares its width with the orange rectangle (x), has a height of y-3, and has an area 3* as big as the orange rectangle's one I noticed that y(x-3)=3(3y). From there, I simplified it to xy-3y=9y, divided each side by y, and added 3 to both sides, wich told me that x=12.
A whole world would have been way better at math (and consequently a lot of other fields that use math, namely, almost everything) if you were their math teacher.
1. Call the side length of the whole square x. 2. Call the unknown side length of the orange rectangle y. 3. The area of the orange (and every other) rectangle is 3y. 4. The area of all five rectangles together (of the square) is 5×3y=15y. 5. The area of the square is also x^2. 6. We have that x^2=15y. 7. The blue rectangle has side lengths x and (x-y). 8. The area of the blue rectangle is also x(x-y). 9. We have x(x-y)=3y. 10. Expand (9) to get x^2-xy=3y. 11. Substitute (6) in (10) to get 15y-xy=3y. 12. Rearrange (11) to get 12y=xy. 13. Since x is at least 3 and x^2=15y, y cannot be 0. 14. Divide both side of (12) by y to get x=12. 15. The area of the square is x^2=12^2=144.
Cool! Here's the way I did it: x is the unknown side of the orange rectangle and L is the side of the square. A(Orange) = 3x A(Square) = L^2 = 5 * Orange = 15x A(Blue) = L^2 - Lx A(Blue) = A(Orange) 3x = L^2 - Lx 3x = 15x - Lx L = 15 - 3 = 12 A(Square) = L^2 = 12^2 = 144
Just looking at the thumbnail, the yellow part is MUCH larger than the orange, green and pink parts. It may also be larger than the blue part. But of course the picture is just a general idea, not precise dimensions. Anyway, nice problem.
damn i did it using a much harder method label the top side of the orange rectangle x and the top side of the blue rectangle y so that the squares sidelength is x+y the green and pink rectangle are identical so well group them up, this newly formed rectangle has the same height as the yellow rectangle which means its width is twice as much as the yellow rectangles width, label them z and 2z, 3z=x which means the area of orange is 9z and the area of yellow is z(3z+y-3), since those are equal we get z(3z+y-3)=z(9) 3z+y-3=9 3z+y=12 since the squares sidelength is x+y and x=3z we get that the squares sidelength is 3z+y aka 12 which means the area is 12² or 144
Ill do you one better: marking the width of the orange rectangle as x, the area is 3x, and the area of the square is 15x and every shape is 3x. because blue is also 3x, if we omit it, then the area of the remaining rectangle is 12x. because the width is x, then the height must be 12. because 15x is a square, and one side is 12, then the area is 144.
Figured it out with this (3 + yellow h) * orange w = 4 * (3 * orange width) Get the answer for yellow height, add it with 3 and since it's a square just square that and voila
All colored rectangles have same area. Pink and green have one side that is the same length. Therefore, they have the exact same dimensions. Their height is g and their width is p. This means that yellow is twice the height of green (or pink). Therefore, it must be half the width. Yellow's height is 2g and its width is p/2. Orange's width is yellow's width plus green's width. Since green is twice as wide as yellow, orange is three times as wide as yellow. Orange's width is 3p/2. Again, all colored rectangles have the same area. This means that orange's area, 3(3p/2), equals green's, which is pg. This means that 9p/2=pg. Divide p from both sides, and 9/2=g. Yellow's height is twice that of green's. Green's height is 9/2. This means that yellow's height is 9. One side of the overall square is 3 plus yellow's height. Yellow's height is 9, which means that one side of the overall square is 12. Total area of the square is 12 squared, or 144.
I preferred to use some geometry facts in this one: 1) Pink and green share a length, and since the areas are the same they are congruent and that is the key to the rest 2) if yellow is l tall and w wide, then pink and green are l/2 tall 3) since the areas are shared, pink and green must be 2w wide 4) orange is as wide as yellow and green combined, and is therefore 3w wide 5) the area of orange is 3*3w for 9w 6) since area of yellow = area of orange, lw=9w; thus l=9 7) the square is yellow and orange tall, hence the side length is 12 8) if side length is 12, area of the square is 144 ||||
Unfortunately, an explanation like this would have been marked wrong, since graphics are not to scale unless specifically stated or given the right numbers. I lost points throughout advanced algebra because I trusted the drawings
@@fizzlock: Their answer doesn't reply on graphics being to scale or with lines exactly drawn in or anything like that; the two rectangles in question necessarily must share the same width due to being bounded on each side by rectangles that span the line where they separate. Everything stated in the above answer is completely possible to derive from the geometry here. That being said, you can make it much easier: yellow, green, and pink are as wide as orange, and since the area of each rectangle is equal, we thus know that the combined height of those three must be equal to three times the height of orange; since this combined height makes up the rest of the height of the entire square, we know the height must be 3 + 3 * 3 = 12.
I argued that yellow+pink+green= 3×Orange( in therms of area ) That means that the height of Yellow+pink+green=3xOranges height cause they have the same total width. Yellow height must be 3x3=9. So the left side of the square =3+9=12. Area of square is height squared so 12x12=144
my solution : consider yellow's height as x, then the green/pink (choose one) as 1/2x. Now consider the yellow length as y and green as z. Compare them both. We get xy = xz/2. We get 2y = z. So now we get the equation for orange where 3 x 3y = xy. we get x = 9. Just add it with the orange height so we get 12 for the sides.
Speedrun version: call the short side of the blue rectangle Y, and the side of the whole square x. Because all recangles have the same area, the orange rectangle which is equal to 3(x-y) is the same as the blue rectangle, which is xy. So xy=3(x-y). Multiply all terms by 4 to get 4xy=12(x-y) Because all rectangles are equal size, the four left rectangles are equal to 4xy. Their area can also be expressed as x(x-y) because we have already defined y as the short side of the blue rectangle, and that length is subtracted from x to get the short side length of the cumulative 4 other rectangles, and the long side of the four left rectangles cumulattively is just x. so 12(x-y)=4xy=x(x-y). Therefore 12(x-y)=x(x-y). Everything but a single x and the number twelve cancels out, and you're left with x=12. 12 squared is 144. It took me 5 lines of algebra and 45 seconds to find this proof, after I solved it a much more complicated way first and then realized DUH.
Actual speedrun version: notice that the three rectangles below the orange one make up one larger rectangle that's exactly as wide as the orange one; since you know that this rectangle has three times of the area as the orange one, you immediately know that its height must be three times that of the orange one, and voilà, you now almost instantly know the height of the entire square is 3 + 9 = 12, which squared is 144.
Orange = 3 × a =3a All have the same area so.... Orange + Yellow + Green + Pink = 3a + 3a + 3a + 3a = 12a Same width "a" as Orange. Left side is 12 so because this is a square. 12×12=144
Yellow, green and pink all have the same area as orange. So together, they have 3x the area of orange. But they form a rectangle the same width as orange, so that rectangle must have 3x the height. Now we already know the square has area (3 + 3·3)² = 144. But we can figure out more! Blue has full height and 1/5 of the area, so its width is 12/5 and thus orange has width 48/5. Yellow has the same height as green+pink but half the area, so it has half the width. Thus yellow has width 48/5/3 = 16/15 and green and pink have width 32/5 and height 9/2 (half of yellow).
Haven’t watched yet but you can solve this with just the blue and orange rectangle. Suppose x is the sidelength of the square Now we have x^2/5 = 12x/5 from the side lengths of the blue and orange rectangle. From this we multiply by 5 and divided by x to get x = 12 so the area is 144
My logic was if the blue rectangle is the height of the square and is 1/5th the width of the square, then the orange rectangle which is 4/5ths the base of the square will be 1/4 the height of the square. We know the orange rectangles height is 3 se we just multiply by 4 to get 12 and then square 12 to get 144. Solved in 2 mins!
The orange area is the same width as the yellow+green+pink area, so it must be 1/3 the height. So the yellow area is height 9, which makes the left side 12, which makes the area 144.
here is the reasoning for those who don't get it. Orange area (O) = 1/5x Yellow + Pink + Green area (Y) = 3/5x 1/5x = (3/5x)÷3 Therefore O = Y÷3 Width of Orange area (W) = Width of Yellow + Green + Pink area (W) Height of O = 3 Height of Y = h O = (3•W) Y = 3•O = (h•W) 3(3•W) = 9•W = (h•W) (9•W)÷W = (h•W)÷W 9 = h (Height of Yellow + Pink + Green area)
@@IsaacRoz Because pink + yellow + green = 3 orange. They all share the same width so the only way to get 3 orange would be to stack them on each other with their widths all equaling the width of the original orange. That gives you 4 oranges stacked on the left each with a height of 3. 3 x 4 = 12 which is the length of one side. 12 x 12 (since it is a square) = 144 units.
So overcomplicated. As others pointed out Yellow + Green + Pink + Orange are 4 times bigger than Orange alone but same width, so height is 4 x 3 = 12. So square is 12 x 12 = 144.
actually you tried hard ways there is simple one. if you say x to upper line of orange one its area will be 3x becouse of the all areas same yellow, pink and green ones Total area will be 9x under line of these tree have to be x becquse same with oranges upper line in the end we know bottom x and area 9x if we divide them we will found other side of rectangle so yellow ones left corner is 9 and orenge ones is 3 we found one corner of square area of square is 12x12 144 simple
That was bizarre. I watched the whole video waiting for the point where you say "But you don't have to do all that. In fact you can ignore the blue rectangle entirely. You only need to look at the other four rectangles." Instead you just said "And this is the answer." And that was the end of the video!
I know an easier way Like, ignore the blue U know that Yellow+pink+green=3Orange Since the length of this 3 color rectangular is the same we have in orange, the height of this 3 colored rectangle is 3 times the height of the orange (3x3=9) (A=3O, A=c×l, O=h×l >> 3hl=cl, c=3h ) So in total, the Side of the square is 9+3=12, and the area 144
I just let green side be a And then 3a=x²/5 So, a=x²/15 We know, Area of blue rectangle will be x(x-a)=x²/5 Subsituting value of a will get us x(x-x²/15)=x²/5 15x²-x³=3x² 12x²=x³ x=12 Love from a person who also loves maths ❤❤
Green and Pink have same A and w, so they must also have same h. Call that h, y. Yellow's h is 2 of those, so 2y. And it's w must be half of Pink's. So call Yellow's w x and Pink's w 2x. Now Orange's w is 3x and it's A is 9x. So Yellow's h is 9x/x or 9 and the side is 12. 12 squared is 144. No square roots necessary. 😂
Niccolo Machiavelli said there were three kinds of intelligence... 3. People who can't understand what they're taught. 2. People who understand what they're taught. 1. People who understand things for themselves without being taught. Andy clearly is the first kind of intelligent. The one who understands things for himself without being taught. I wanna know how he does it. I wonder what his SAT scores are, or his IQ scores. Or his CAIT scores? He should do the CAIT, I hear it's free.
You could turn yellow pink and green into the same shape as orange. Now you have 4 orange ones stacked. 4 times 3 is 12. No calculations, not even looking at blue 😂
It was easy. If you call the largest part of yellow rectangle as Y and the largest part of orange rectangle as X, you have: (Y * X)/3 = 3X Y * X = 9X Knowing X != 0: Y = 9 We know the side of the square is equal to Y + 3, so the side of square is equal to 12 and the area is 144 area unit.
Holy crap! Why did you go around the moon to get to the end of the block? Since we know that all rectangles are the same size, then we know that pink + yellow + green = 3 x orange. Since we know that pink + yellow + green all share the same width then the only way to get 3 x orange would be to lay 3 orange rectangles side by side to cover the area of pink + yellow + green. That gives you 4 orange rectangles stacked on the left. 4 x 3 =12. Since it is a square 12 x 12 gives you 144 square units. Simple enough?
Could you do a video of the bisecting angle of a pyramid corner? One day, I would like to make a pyramid out of plywood, but don't know at what angle to cut it. It's not 45 degrees. I just saw a video where the degree was 54.6 degrees, but wonder how they got that. Also wonder how the height of the pyramid would affect the angle. Actually, could you also figure out the angle for a 4 sided pyramid as well? Here's the video: czcams.com/video/Y6B-vTxjsVk/video.html
The formula you need (in degrees) is chamfer angle = arcsin[cos(180°/n)/cos(α/2)] (where arcsin means the same as sin⁻¹). This is the chamfer angle for the inclined sides of the triangular faces of a pyramid with a regular n-sided polygon base and n triangular sides, each of which is an isosceles triangle with apex angle α. Note that the chamfer angle is simply half the angle between the adjacent faces. For example, in the case of a pyramid with a square base and equilateral triangular sides in the video you linked to, we have n=4 and α=60°, so we get the chamfer angle arcsin[cos(180°/4)/cos(60°/2)] =arcsin[cos(45°)/cos(30°)] =arcsin[(√2/2)/(√3/2)] =arcsin(√2/√3) =arcsin(√(2/3)) =arcsin(0.816496...) =54.7356...° (I did the final calculations using the Calculator app on my Android phone). Although I worked out this formula myself, it must be very well known, as it can be used, for example, to calculate the angle between adjacent sides of any regular polyhedron. Note that to use the formula we don't need the joining faces to be triangles; it applies to n corners of angle α meeting "regularly" at a point, as we can cut off a pyramid with a regular polygonal base and isosceles triangular faces close to the point. As the angle between adjacent faces is twice the chamfer angle we have (in degrees) θ=2arcsin[cos(180°/n)/cos(α/2)] where θ is this angle. For example, for a cube, we have three faces of angle 90° meeting at each vertex, so n=3 and α=90°, and θ=2arcsin[cos(180°/3)/cos(90°/2)] =2arcsin[cos(60°)/cos(45°)] =2arcsin[(1/2)/(1/√2)] =2arcsin(√2/2) =2×45° =90°, as expected. --- EDIT: Here is an explanation of how I got the formula (not easy to write or follow without diagrams!) Let ΔAOB and ΔBOC be two adjacent isosceles triangular faces of the pyramid with apex O, where OA=OB=OC=1 (the equal sides of the isosceles triangles),
@@MichaelRothwell1 Wow, thank you. I am decent in math, but would have never gotten that on my own. I don't know when I'll make a pyramid ever, but I did make a tapered base in uni for a friend and couldn't figure it out then. Another friend told me an angle and I tried it and it was too shallow, but we used a lot of filler and got it to work. Since then, I've always wondered what the math behind it was. Thank again!
@@mrtienpham I've now updated my first reply to include a derivation of the formula. It will be easier to follow if you use pencil and paper to draw the diagrams as you go along. Good luck!
There's an even simpler solution:
1) There are Yellow, Green and Pink rectangles under the Orange one. That means, that the combined rectangle YGP (containing the Yellow, Green and Pink rectangles) have the same width as the Orange one.
2) The area of the YGP rectangle is 3 times greater than the area of the Orange rectangle (since it's a composite of 3 rectangles of same area)
3) Since Orange and YGP have the same width and the are ratio of 1/3, that means that their height ratio is also 1/3, resulting in the YGP height is 9
4) The side of the Square is 3 + 9 = 12, and the area is 144
I did it the same way, it was a lot too easy
@@luisfagnerdecarvalhosilva9765 i also did it this way. thought i'd need pen/paper but did it mentally in about 30seconds.
When you have fancy tools, everything looks like a fancy job..
But when you want to get the job done, the simplest solution is best.
This is the best solution.
@@peterpan408what? Lol
@@therealdoc All of the squares have the same area. If you get rid of the blue section you have the combined area of the Yellow Green and Pink sections is 3 times that of the orange section with the same width, therefore they must have height 3 times greater than orange to have an area 3 times greater. This means their height must be 9 making the combined height of the larger square 9+3=12 and then 12*12 is 144.
My favorite videos to watch on CZcams and it feels weird but “how exciting” it is to find the solution to these problems.
Yeah
What I love to watch here is the process to find the solution
No bs, no weird promotions, no trying to act important, and so on.
Just plain old crystal clear problem solving.
This is oddly satisfying hahaha!
These are awesome! Keep them coming. It has also been a really good refresher course.
andy thank you for smiling as you make these videos; truly wonderful and interesting to watch!
I really appreciate these videos. Helps keep out problem solving skills sharp, applicable to many real world applications
i love these videos, they keep me in practice during the holidays!!!
This problem have different ways to solve it, how exiting!
My solution was Yellow+Green+Pink = 3x Orange, so I must be able to place 3 other Oranges bellow the one given, so I am left with 4x orange and a blue. each orange has the side of 3, so 4x3 = 12 = x ...
elegant
The exact problem was in my maths competition for 8th graders that was in february this year, love to see your solution.
i did it an easier way. The yellow, green, and pink rectangles make a big rectangle with unknown lengths. Lets call those lengths x and y. The area of that rectangle is then xy. Area of orange rectangle = 3x. we know that the area of the orange rectange is 1/3 times the area of the combined rectangle. therefore 3(3x) = xy. 9x = xy. y = 9. Therefore the length of one side of the square is 3 + 9 = 12. 12 * 12 = 144.
Great lesson =)
"Let's put a box around it." i love you andy
I love how you only needed to analize 2 squares out of 5 to get the answer
Yeah! Ended up being much easier than I you'd think at first glance, I was surprised
You're a legend bro 💯🔥
I like your videos - straight forward, not wasting time rewriting equations ad infinitum to add minutes.
I also solved this the simpler way it's been discussed in comments, but also have a small problem with the presentation. Put simply, we had to assume the smaller portions were all rectangles, even though it wasn't stated.
Thanks Andy
This is mind-blowing
Thanks for the solution❤❤
Exactly how I solved it. How, exciting.
Also, Andy, what program do you use to make these videos?
I love problems where there's many different ways to solve. Here's the fastest I found:
Let "A" be the area of a colored rectangle. The bottom side of the orange rectangle must be A/3. Then, let "x" be the height of the yellow rectangle.
The bottom of the orange rectangle and left side of the yellow rectangle can be the perpendicular sides of a new rectangle, with area x*A/3. This rectangle is the same as the yellow, green, and pink rectangles combined, so we know it also has an area of 3A.
Ax/3=3A, therefore x (the height of the yellow rectangle) is 9.
Simple arithmetic from there, 9+3=12 and 12²=144.
I had a lot of fun calculating the sides of each rectangle aswell
Simpler Solution:
Area of orange box = 3x, where x is the unknown side of the orange box
Area of all boxes but the blue = x * the side length of the whole square
So the side length of the whole square = the area of all but blue divided by the top side of the orange box, x: 12x / x = 12
Square this to get the area of the whole square: 12^2 = 144
Even simpler: pink green and yellow the same side as orange but have 3 times the length so the area is (3*4)² = 144
exactly how I did this
There was no point in calculating shorter edge of the blue rect. It had to be 1/5 of x.
My solution was a lot more complicated, but super happy I got it!
Another solution
Green and pink are equal, because areas and horizontal sides also are. So as yellow vertical is 2*(green or pink) vertical, and the areas are the same, yellow horizontal its ½ green vertical
So, orange vertical is 3* yellow horizontall. Then, as areas are the same, 3*orangeV = 3*3*yellowH = yellowH*yellowV, so yellowV is 9. So then, the side of the square is 3+9=12 length
I thought you forgot to put a box around the answer, nice one!
One man's 'Exciting' is another man's 'Relaxing'
How elegant!
I’m imagining this is what Andy does on a Friday night at Midnight. 🎉
one day i will solve one of these without having to watch you do it, today is not that day, but one day
Would this be solvable using trigonometric functions give you have "opposite"?
you made me remember I love math
Since there are three rectangles under the green one, the length of the yellow one must be 3 * 3 = 9.
So the side of the square must be 3 + 9 = 12, and, therefore, its area 12² = 144 square units.
I solved it a different, and I think easier way: Ignore the blue rectangle. If you re-shape the orange, green and pink rectangles so they're the same shape as the orange one while preserving their area, all four will be exactly the same as the orange one... therefore there's 4 that can fit into the whole square minus the blue rectangle, therefore the side length is 4 x 3 = 12
And that’s the way I did it. Simpler and quicker
Another algebra-free way.
1. Green and pink are the same width so, because they're the same area, they're also the same height as each other, ie they're identical.
2. Yellow is twice as high as either green or pink so is half as wide as them.
3. Orange is three times the width of yellow because it's as wide as yellow and green, and green is twice as wide as yellow.
4. If orange is three times as wide as yellow, it's a third of the height, so yellow has height 9, and the square height 12.
5. You can use the fact we know that it's square to get the answer, 144.
I got there through another way.
First, I let the width of the orange rectangle be x, and the side of the big square be y. Then, since the rectangle formed by yellow, green and pink shares its width with the orange rectangle (x), has a height of y-3, and has an area 3* as big as the orange rectangle's one I noticed that y(x-3)=3(3y). From there, I simplified it to xy-3y=9y, divided each side by y, and added 3 to both sides, wich told me that x=12.
A whole world would have been way better at math (and consequently a lot of other fields that use math, namely, almost everything) if you were their math teacher.
1. Call the side length of the whole square x.
2. Call the unknown side length of the orange rectangle y.
3. The area of the orange (and every other) rectangle is 3y.
4. The area of all five rectangles together (of the square) is 5×3y=15y.
5. The area of the square is also x^2.
6. We have that x^2=15y.
7. The blue rectangle has side lengths x and (x-y).
8. The area of the blue rectangle is also x(x-y).
9. We have x(x-y)=3y.
10. Expand (9) to get x^2-xy=3y.
11. Substitute (6) in (10) to get 15y-xy=3y.
12. Rearrange (11) to get 12y=xy.
13. Since x is at least 3 and x^2=15y, y cannot be 0.
14. Divide both side of (12) by y to get x=12.
15. The area of the square is x^2=12^2=144.
Cool!
Here's the way I did it:
x is the unknown side of the orange rectangle and L is the side of the square.
A(Orange) = 3x
A(Square) = L^2 = 5 * Orange = 15x
A(Blue) = L^2 - Lx
A(Blue) = A(Orange)
3x = L^2 - Lx
3x = 15x - Lx
L = 15 - 3 = 12
A(Square) = L^2 = 12^2 = 144
x: area of square
y : side of square
z : unknown side of orange rectangle
5x = y*y
3x = (y-3)*z
x = z*3
9=y-3
y =12
area = 12*12
How exciting!
Just looking at the thumbnail, the yellow part is MUCH larger than the orange, green and pink parts. It may also be larger than the blue part. But of course the picture is just a general idea, not precise dimensions. Anyway, nice problem.
damn i did it using a much harder method
label the top side of the orange rectangle x and the top side of the blue rectangle y so that the squares sidelength is x+y
the green and pink rectangle are identical so well group them up, this newly formed rectangle has the same height as the yellow rectangle which means its width is twice as much as the yellow rectangles width, label them z and 2z, 3z=x which means the area of orange is 9z and the area of yellow is z(3z+y-3), since those are equal we get
z(3z+y-3)=z(9)
3z+y-3=9
3z+y=12
since the squares sidelength is x+y and x=3z we get that the squares sidelength is 3z+y aka 12 which means the area is 12² or 144
damn i was so perplexed by 3 i didnt consider the blue square
Ill do you one better: marking the width of the orange rectangle as x, the area is 3x, and the area of the square is 15x and every shape is 3x. because blue is also 3x, if we omit it, then the area of the remaining rectangle is 12x. because the width is x, then the height must be 12. because 15x is a square, and one side is 12, then the area is 144.
I know i can do this, but i got lazy thinking after he finished counting down so i just let Andy do it.
Figured it out with this
(3 + yellow h) * orange w = 4 * (3 * orange width)
Get the answer for yellow height, add it with 3 and since it's a square just square that and voila
All colored rectangles have same area.
Pink and green have one side that is the same length. Therefore, they have the exact same dimensions. Their height is g and their width is p.
This means that yellow is twice the height of green (or pink). Therefore, it must be half the width. Yellow's height is 2g and its width is p/2.
Orange's width is yellow's width plus green's width. Since green is twice as wide as yellow, orange is three times as wide as yellow. Orange's width is 3p/2.
Again, all colored rectangles have the same area. This means that orange's area, 3(3p/2), equals green's, which is pg. This means that 9p/2=pg. Divide p from both sides, and 9/2=g.
Yellow's height is twice that of green's. Green's height is 9/2. This means that yellow's height is 9.
One side of the overall square is 3 plus yellow's height. Yellow's height is 9, which means that one side of the overall square is 12.
Total area of the square is 12 squared, or 144.
Nice and elegant!
area(rectangle(yellow+green+pink))=3*area(orange).
width(rectangle(yellow+green+pink))=width(orange).
height(rectangle(yellow+green+pink))=3*height(orange)=3*3=9.
height(whole square)=height(orange)+height(rectangle(yellow+green+pink))=3+9=12=width(whole square).
area(whole square)=height(whole square)*width(whole square)=12*12=144.
Suppose we have 3 biggest possible circle in a big circle. If the radius of the small circle radius is 5, then what's the area of the big circle?
en.m.wikipedia.org/wiki/Circle_packing_in_a_circle
@@tiagoschmidt Thx for ur link, but I hope Andy can teach out how to do it
I preferred to use some geometry facts in this one:
1) Pink and green share a length, and since the areas are the same they are congruent and that is the key to the rest
2) if yellow is l tall and w wide, then pink and green are l/2 tall
3) since the areas are shared, pink and green must be 2w wide
4) orange is as wide as yellow and green combined, and is therefore 3w wide
5) the area of orange is 3*3w for 9w
6) since area of yellow = area of orange, lw=9w; thus l=9
7) the square is yellow and orange tall, hence the side length is 12
8) if side length is 12, area of the square is 144
||||
Unfortunately, an explanation like this would have been marked wrong, since graphics are not to scale unless specifically stated or given the right numbers. I lost points throughout advanced algebra because I trusted the drawings
@@fizzlock:
Their answer doesn't reply on graphics being to scale or with lines exactly drawn in or anything like that; the two rectangles in question necessarily must share the same width due to being bounded on each side by rectangles that span the line where they separate. Everything stated in the above answer is completely possible to derive from the geometry here.
That being said, you can make it much easier: yellow, green, and pink are as wide as orange, and since the area of each rectangle is equal, we thus know that the combined height of those three must be equal to three times the height of orange; since this combined height makes up the rest of the height of the entire square, we know the height must be 3 + 3 * 3 = 12.
I argued that yellow+pink+green= 3×Orange( in therms of area )
That means that the height of Yellow+pink+green=3xOranges height cause they have the same total width.
Yellow height must be 3x3=9. So the left side of the square =3+9=12.
Area of square is height squared so 12x12=144
my solution : consider yellow's height as x, then the green/pink (choose one) as 1/2x. Now consider the yellow length as y and green as z. Compare them both. We get xy = xz/2. We get 2y = z. So now we get the equation for orange where 3 x 3y = xy. we get x = 9. Just add it with the orange height so we get 12 for the sides.
How exciting.
Interesting that you didn’t even need the given information of 3 in the top left of the problem. How exciting!
But he did, at 1:31. It’s necessary. Suppose, for example, that the shapes remained the same but the 3 becomes 10. The area is then 1600.
Speedrun version: call the short side of the blue rectangle Y, and the side of the whole square x. Because all recangles have the same area, the orange rectangle which is equal to 3(x-y) is the same as the blue rectangle, which is xy. So xy=3(x-y). Multiply all terms by 4 to get 4xy=12(x-y)
Because all rectangles are equal size, the four left rectangles are equal to 4xy. Their area can also be expressed as x(x-y) because we have already defined y as the short side of the blue rectangle, and that length is subtracted from x to get the short side length of the cumulative 4 other rectangles, and the long side of the four left rectangles cumulattively is just x.
so 12(x-y)=4xy=x(x-y). Therefore 12(x-y)=x(x-y). Everything but a single x and the number twelve cancels out, and you're left with x=12. 12 squared is 144.
It took me 5 lines of algebra and 45 seconds to find this proof, after I solved it a much more complicated way first and then realized DUH.
Actual speedrun version: notice that the three rectangles below the orange one make up one larger rectangle that's exactly as wide as the orange one; since you know that this rectangle has three times of the area as the orange one, you immediately know that its height must be three times that of the orange one, and voilà, you now almost instantly know the height of the entire square is 3 + 9 = 12, which squared is 144.
How exciting ☘️
Cool video but why is the image not accurate
Orange = 3 × a =3a
All have the same area so....
Orange + Yellow + Green + Pink = 3a + 3a + 3a + 3a = 12a
Same width "a" as Orange. Left side is 12 so because this is a square.
12×12=144
The best solution, imho!
Ayy midnight vid this time
Yellow, green and pink all have the same area as orange. So together, they have 3x the area of orange. But they form a rectangle the same width as orange, so that rectangle must have 3x the height. Now we already know the square has area (3 + 3·3)² = 144.
But we can figure out more! Blue has full height and 1/5 of the area, so its width is 12/5 and thus orange has width 48/5. Yellow has the same height as green+pink but half the area, so it has half the width. Thus yellow has width 48/5/3 = 16/15 and green and pink have width 32/5 and height 9/2 (half of yellow).
Really going for extra credits here, eh?
Haven’t watched yet but you can solve this with just the blue and orange rectangle. Suppose x is the sidelength of the square
Now we have x^2/5 = 12x/5 from the side lengths of the blue and orange rectangle. From this we multiply by 5 and divided by x to get x = 12 so the area is 144
Thats exactly how you did it lol
I'm going to solve it in fewer steps
1) Take out a tape measurer
2) Measure the sides
3) Multiply them together
me procrastinating on yt to watch these videos.
this guy ❤
My logic was if the blue rectangle is the height of the square and is 1/5th the width of the square, then the orange rectangle which is 4/5ths the base of the square will be 1/4 the height of the square. We know the orange rectangles height is 3 se we just multiply by 4 to get 12 and then square 12 to get 144. Solved in 2 mins!
Why do you multiply both sides by 5/x
How exciting🗣️🗣️🔥
When I saw the word Engineering, I said those sides look to be about 9 so 81
How exciting
The orange area is the same width as the yellow+green+pink area, so it must be 1/3 the height. So the yellow area is height 9, which makes the left side 12, which makes the area 144.
how can you confirm the height of yellow is 9 based on 1:3 ratio? it's not stated in the question, nor during the calculation in the video.
they can be different, for this it works obviously but for others its best not to make a habit of trying to eyeball
Blue is irrelevant here so imagen it gone
here is the reasoning for those who don't get it.
Orange area (O) = 1/5x
Yellow + Pink + Green area (Y) = 3/5x
1/5x = (3/5x)÷3
Therefore O = Y÷3
Width of Orange area (W) = Width of Yellow + Green + Pink area (W)
Height of O = 3
Height of Y = h
O = (3•W)
Y = 3•O = (h•W)
3(3•W) = 9•W = (h•W)
(9•W)÷W = (h•W)÷W
9 = h (Height of Yellow + Pink + Green area)
@@IsaacRoz Because pink + yellow + green = 3 orange. They all share the same width so the only way to get 3 orange would be to stack them on each other with their widths all equaling the width of the original orange. That gives you 4 oranges stacked on the left each with a height of 3. 3 x 4 = 12 which is the length of one side. 12 x 12 (since it is a square) = 144 units.
3 * orange = yellow + green + pink => yellow left side is 9 => the cube side is 12 => the area is 144
I made a drawing of this problem to scale
So overcomplicated. As others pointed out Yellow + Green + Pink + Orange are 4 times bigger than Orange alone but same width, so height is 4 x 3 = 12. So square is 12 x 12 = 144.
actually you tried hard ways there is simple one. if you say x to upper line of orange one its area will be 3x becouse of the all areas same yellow, pink and green ones Total area will be 9x under line of these tree have to be x becquse same with oranges upper line in the end we know bottom x and area 9x if we divide them we will found other side of rectangle so yellow ones left corner is 9 and orenge ones is 3 we found one corner of square area of square is 12x12 144 simple
i tried using the cuadratic formula and got 36^2 idek how
That was bizarre. I watched the whole video waiting for the point where you say "But you don't have to do all that. In fact you can ignore the blue rectangle entirely. You only need to look at the other four rectangles."
Instead you just said "And this is the answer." And that was the end of the video!
I know an easier way
Like, ignore the blue
U know that Yellow+pink+green=3Orange
Since the length of this 3 color rectangular is the same we have in orange, the height of this 3 colored rectangle is 3 times the height of the orange (3x3=9)
(A=3O, A=c×l, O=h×l >> 3hl=cl, c=3h )
So in total, the Side of the square is 9+3=12, and the area 144
personally i found out how long the side of the square in pixels and discoverde 300. so its 9x9 = 81. guess it didnt work tho xD
I tried it
And ends up with 4 different variable,
I laughed at myself After watching this
I just let green side be a
And then 3a=x²/5
So, a=x²/15
We know,
Area of blue rectangle will be x(x-a)=x²/5
Subsituting value of a will get us
x(x-x²/15)=x²/5
15x²-x³=3x²
12x²=x³
x=12
Love from a person who also loves maths ❤❤
Why did x/5 suddenly turn into 1/5 at 1:04?
Exiting how.
Green and Pink have same A and w, so they must also have same h. Call that h, y. Yellow's h is 2 of those, so 2y. And it's w must be half of Pink's. So call Yellow's w x and Pink's w 2x. Now Orange's w is 3x and it's A is 9x. So Yellow's h is 9x/x or 9 and the side is 12. 12 squared is 144. No square roots necessary. 😂
Spent an hour on this before realizing the area of a square is (sidelength)^2 and not (sidelength)*2.... guess I need to go back to grade school
…or just measure the “3” side
Niccolo Machiavelli said there were three kinds of intelligence...
3. People who can't understand what they're taught.
2. People who understand what they're taught.
1. People who understand things for themselves without being taught.
Andy clearly is the first kind of intelligent. The one who understands things for himself without being taught. I wanna know how he does it. I wonder what his SAT scores are, or his IQ scores. Or his CAIT scores? He should do the CAIT, I hear it's free.
I think I might be number 4 😂
@@billycox475And I think I might be a solid 2 1/2.
You could turn yellow pink and green into the same shape as orange. Now you have 4 orange ones stacked. 4 times 3 is 12. No calculations, not even looking at blue 😂
you an be high school math teacher
Orange = 3 x 9.6
Blue = 12 x 2.4
Yellow = 9 x 3.2
Green = 6.4 x 4.5
Pink = 6.4 x 4.5
It was easy.
If you call the largest part of yellow rectangle as Y and the largest part of orange rectangle as X, you have:
(Y * X)/3 = 3X
Y * X = 9X
Knowing X != 0:
Y = 9
We know the side of the square is equal to Y + 3, so the side of square is equal to 12 and the area is 144 area unit.
Cool
It feels weird, because blue is equal to 12 x 2.4? That is lesser than 36
It is, but why is 36 significant?
🎉❤
How exckxckxkckxiting
I paused it and it's 5cm*5cm haha (I measured it)
X = 3(people who have never been in my kitchen)
Holy crap! Why did you go around the moon to get to the end of the block? Since we know that all rectangles are the same size, then we know that pink + yellow + green = 3 x orange. Since we know that pink + yellow + green all share the same width then the only way to get 3 x orange would be to lay 3 orange rectangles side by side to cover the area of pink + yellow + green. That gives you 4 orange rectangles stacked on the left. 4 x 3 =12. Since it is a square 12 x 12 gives you 144 square units. Simple enough?
Could you do a video of the bisecting angle of a pyramid corner? One day, I would like to make a pyramid out of plywood, but don't know at what angle to cut it. It's not 45 degrees. I just saw a video where the degree was 54.6 degrees, but wonder how they got that. Also wonder how the height of the pyramid would affect the angle.
Actually, could you also figure out the angle for a 4 sided pyramid as well?
Here's the video: czcams.com/video/Y6B-vTxjsVk/video.html
The formula you need (in degrees) is
chamfer angle
= arcsin[cos(180°/n)/cos(α/2)]
(where arcsin means the same as sin⁻¹).
This is the chamfer angle for the inclined sides of the triangular faces of a pyramid with a regular n-sided polygon base and n triangular sides, each of which is an isosceles triangle with apex angle α.
Note that the chamfer angle is simply half the angle between the adjacent faces.
For example, in the case of a pyramid with a square base and equilateral triangular sides in the video you linked to, we have n=4 and α=60°, so we get the chamfer angle
arcsin[cos(180°/4)/cos(60°/2)]
=arcsin[cos(45°)/cos(30°)]
=arcsin[(√2/2)/(√3/2)]
=arcsin(√2/√3)
=arcsin(√(2/3))
=arcsin(0.816496...)
=54.7356...°
(I did the final calculations using the Calculator app on my Android phone).
Although I worked out this formula myself, it must be very well known, as it can be used, for example, to calculate the angle between adjacent sides of any regular polyhedron.
Note that to use the formula we don't need the joining faces to be triangles; it applies to n corners of angle α meeting "regularly" at a point, as we can cut off a pyramid with a regular polygonal base and isosceles triangular faces close to the point.
As the angle between adjacent faces is twice the chamfer angle we have (in degrees)
θ=2arcsin[cos(180°/n)/cos(α/2)]
where θ is this angle.
For example, for a cube, we have three faces of angle 90° meeting at each vertex, so n=3 and α=90°, and
θ=2arcsin[cos(180°/3)/cos(90°/2)]
=2arcsin[cos(60°)/cos(45°)]
=2arcsin[(1/2)/(1/√2)]
=2arcsin(√2/2)
=2×45°
=90°, as expected.
---
EDIT:
Here is an explanation of how I got the formula (not easy to write or follow without diagrams!)
Let ΔAOB and ΔBOC be two adjacent isosceles triangular faces of the pyramid with apex O, where OA=OB=OC=1 (the equal sides of the isosceles triangles),
@@MichaelRothwell1 Wow, thank you. I am decent in math, but would have never gotten that on my own.
I don't know when I'll make a pyramid ever, but I did make a tapered base in uni for a friend and couldn't figure it out then. Another friend told me an angle and I tried it and it was too shallow, but we used a lot of filler and got it to work. Since then, I've always wondered what the math behind it was.
Thank again!
@@mrtienpham you're welcome! Nice problem!
@@mrtienpham I've now updated my first reply to include a derivation of the formula. It will be easier to follow if you use pencil and paper to draw the diagrams as you go along. Good luck!