Yeah, the shortcut is to look at the face and realize the shaded area is the larger upper triangle's area (12x16)/2 minus the smaller upper triangle's area (6x8)/2. That's 72. Multiply by the depth 50, and you have the answer 3600. Nevertheless, your video was fun to watch! Thanks for the great content.
Even shorter: the way to divide a triangle (in this case the whole triangle consisting of the named trapezoid and the small corner triangle) in 4 same triangles of 4 same areas is to connect all midpoints, and by seeing this you know that the trapezoid has exactly 3x the area of the small triangle. This works for ANY triangle. Thus: V=bxh=3x(8x6)/2*50=3600
Like the fact that you’re solving very simple math problems, for some reason it’s very entertaining, plus it’s good for the brain and keeping logic alive.
I not only made it to the end, but replayed the latter portion calculating h. A long video for this channel is a third the length or shorter compared to math videos elsewhere. I appreciate your method of explanation and brisk pace.
You need to calculate the surface of that is facing us and multiple by 50m. And to do that you start by drawing the smallest rectangle than includes the surface we need. And it its a: (8+8)x(6+6)=192 Than you calculate half of it since the diameter is one of the surface sides: 192/2 = 96 Than you calculate that small triangle on the top left: (8x6)/2 = 24 You add the two triangles: 96+24= 120 Than you substracte the the surface of the two triangles from the rectangle: 192-120=72 Now, you multiply the surface by the depth: 72x50= 3600m³ Sorry, I hope it made sense. My main language isn't English, I also calculated all that in my head.
due to the uniformity of the slices, and the even number of slices, you can reorder the slices to have equal 1/3 volume (slices numbered from left to right): 1+4=2+5=3+6. since slice 2=5 your formula is slice 2 = (18*24*50/3)/2 or 3600
If you pair the long unshaded trapezoids with the triangles, and pair the identical shaded/unshaded trapezoids, you can form a parallelogram where each pair is 1/3rd the total area, thus the shaded trapezoid is 1/6th the area of the 24x18 rectangle.
The really hard way.. simply should be the triangle section made from the top 2 pieces, minus the top piece. (((8+8)*(6+6))/2)*50 - (((8*6)/2)*50) = 16*12*25 - 48*25 = 4800 - 1200 = 3600
i’m proud of myself for remembering the 3-4-5 special triangles and figured out b1 and b2 so much faster. it’s only been like 15 years since i learned that in high school geometry lol
Divide the box into a grid of nine 8x6 boxes, the diagonals halve these boxes, the shaded area is made of three halved boxes, then times the length gives answer (8x6)(3/2)(50) = 3600m^2
Good on you for doing it the hard way. Good on me following along having a history of failing math. I'd love to see something about the middle of triangles. :)
I made it to the end AND understood all of your work. I particularly liked how you isolated all the parts of the original object. How exciting. [ I borrowed that phrase from a math geek on UA-cam 🙄]. Seriously, Andy, great work....as always. 💙👍
I think I see the easy solution, but I'd be bad at proving it. It has to do something with the ratios on the side - like the corner triangle fits 3 times into the middle one, and 5 times into the larger one. Add them up, that 9 triangle where the middle one takes up 3 spaces, so that's 1/3. This is half the rectangle on the side, so I divide it by to to get 1/6. Calculate the whole area, divide by 6.
The triangle area of a triangle is it's base times it's height divided by two, the shaded prism is the area of one triangle minus another. B*H/2 - b*h/2 where B=8+8 H=6+6 b=8 h=6 Once you've found the face area of the prism the length is 50.
9 місяців тому
That's a nice one. But after you found the b1 and b2 I would use trigonometry instead. The angle of the top right corner of the rectangle is the same that the angle of the trapisoid right bottom angle. So, first the rectangle's angle: arctan(24/18) = 53,13° Now the h: h = 6 * sin(53,13) => 6 * 0,8 = 4,8 From now everything the same. I think this is a litle bit simplier.
Good video! I wonder if my geometry students could watch this once as a class and then reproduce what you did. Nothing alone would be new to them, but you do move a bit quicker than they might be able to follow.
No need for Pythagoras, it's a 3:4:5 triangle, and both triangles are similar with scale factor of 2.. I love the triangle area equivalence though, excellent trick for finding the height.
Can you know for sure that the top and bottom of the trapezoid are parallel? The clue just says "prism", but there's nothing in the original diagram to assert that the lines in the base of the shaded area are actually parallel. I don't know if the word "prism" guarantees that does it? e.g. I don't know whether you can just "move the lower labels to the top" (1:06) can you?
Break up the base into 18 triangles of equal area. The shaded part is 3 out of 18, or 1/6 the base area. So the volume is (1/6)(18)(24)(50) which you can do in your head
1:06 isn't that an assumption? It's possible for the top 3 to not be divided into 6/6/6 like the bottom. There's nothing stopping the top segments from varying because they're not connected to the measurements at the bottom. The most I would assume is that the total length of the top is 18m because of the rectangular cross section 🤔
Very nice - made it to the end for sure. How come you chose to do all the math for Pythagorean theorem when you could have pointed out that both triangles followed the 3-4-5 rule?
Andy Math, please solve a OBMEP ( The Brazilian Mathematical Olympiad of Public Schools ) question to be more specific the 6th question of 2023 level 3 phase 1
I will assume that the shaded Area is 1/6 of whole. As we have equal fragments. One is smaler 2nd is average 3rd is the longest. So by averaging, we should have 1/6 of whole intersection field. And just multiply by length Xd, i was right, WHO would gues
Simplifying all the way to 4.8 was an unnecessary step. You had 24/5. In the formula you were dividing by 2 and multiplying by 5. Those 10s would cancel out. It's not always worth it to get all the way to decimal.
I want to do maths like u What are the topics and sub topics I need to learn Assume I know nothing and I'm complete beginner I want to be like u guide me please
Love the dramatic flipping of the triangles.
I don't.
@@powerzxshut up
Yeah, the shortcut is to look at the face and realize the shaded area is the larger upper triangle's area (12x16)/2 minus the smaller upper triangle's area (6x8)/2. That's 72. Multiply by the depth 50, and you have the answer 3600. Nevertheless, your video was fun to watch! Thanks for the great content.
YES, this was one of the easiest problems shown, I finished it fast. How exciting :)
That is probably why the title is I did it the hard way
Even shorter: the way to divide a triangle (in this case the whole triangle consisting of the named trapezoid and the small corner triangle) in 4 same triangles of 4 same areas is to connect all midpoints, and by seeing this you know that the trapezoid has exactly 3x the area of the small triangle. This works for ANY triangle. Thus:
V=bxh=3x(8x6)/2*50=3600
We all made it to the end because of how exciting it was.
i love how he animated it also 3:03 is very cool
Didn't expect that lol.
the taming apart of the other prisms + the triangle flip TRULY made this one exciting. thanks!
Your videos have opened up so many shortcuts and concepts for me with math, I'd love a video on triangle mid segments
Like the fact that you’re solving very simple math problems, for some reason it’s very entertaining, plus it’s good for the brain and keeping logic alive.
I not only made it to the end, but replayed the latter portion calculating h.
A long video for this channel is a third the length or shorter compared to math videos elsewhere.
I appreciate your method of explanation and brisk pace.
You need to calculate the surface of that is facing us and multiple by 50m. And to do that you start by drawing the smallest rectangle than includes the surface we need. And it its a:
(8+8)x(6+6)=192
Than you calculate half of it since the diameter is one of the surface sides:
192/2 = 96
Than you calculate that small triangle on the top left:
(8x6)/2 = 24
You add the two triangles:
96+24= 120
Than you substracte the the surface of the two triangles from the rectangle:
192-120=72
Now, you multiply the surface by the depth:
72x50= 3600m³
Sorry, I hope it made sense. My main language isn't English, I also calculated all that in my head.
due to the uniformity of the slices, and the even number of slices, you can reorder the slices to have equal 1/3 volume (slices numbered from left to right): 1+4=2+5=3+6. since slice 2=5 your formula is slice 2 = (18*24*50/3)/2 or 3600
That prism is unnecessarily huge...
cm would probably be more realistic, but this could be a building or something
You know what else is massive?
If you pair the long unshaded trapezoids with the triangles, and pair the identical shaded/unshaded trapezoids, you can form a parallelogram where each pair is 1/3rd the total area, thus the shaded trapezoid is 1/6th the area of the 24x18 rectangle.
I’m binging your videos. Never got through math in school so I’m learning so much now.
Of course I made it to the end here.
Finding the height of the triangle was so brilliant
It is one of my favorite logical paths. Two things are equal. Change your perspective on one of them. Learn something new.
The really hard way.. simply should be the triangle section made from the top 2 pieces, minus the top piece.
(((8+8)*(6+6))/2)*50 - (((8*6)/2)*50)
= 16*12*25 - 48*25
= 4800 - 1200
= 3600
i’m proud of myself for remembering the 3-4-5 special triangles and figured out b1 and b2 so much faster. it’s only been like 15 years since i learned that in high school geometry lol
Exactly 4:20 minutes, nice
How exciting
I made it to the end, don't worry.
Divide the box into a grid of nine 8x6 boxes, the diagonals halve these boxes, the shaded area is made of three halved boxes, then times the length gives answer (8x6)(3/2)(50) = 3600m^2
Good on you for doing it the hard way. Good on me following along having a history of failing math. I'd love to see something about the middle of triangles. :)
Yeah please make a another video on the mid segment of triangles. Im pretty hooked in!
I made it to the end AND understood all of your work. I particularly liked how you isolated all the parts of the original object. How exciting. [ I borrowed that phrase from a math geek on UA-cam 🙄]. Seriously, Andy, great work....as always. 💙👍
Shortcut method would be subtracting the smaller traingular prism with the bigger triangular prism, but this was fun as well. Thank you
Yes! I made it to the end to check if I got the same answer using the two triangles in the upper left of the face. How exciting.
when simple solutions doesn't satisfy your brain needs
Andy's method is how you get the first 3 problems right and have no time remaining to finish the other 7 on the test before class ends. 😛
I think I see the easy solution, but I'd be bad at proving it. It has to do something with the ratios on the side - like the corner triangle fits 3 times into the middle one, and 5 times into the larger one. Add them up, that 9 triangle where the middle one takes up 3 spaces, so that's 1/3. This is half the rectangle on the side, so I divide it by to to get 1/6. Calculate the whole area, divide by 6.
The triangle area of a triangle is it's base times it's height divided by two, the shaded prism is the area of one triangle minus another.
B*H/2 - b*h/2 where B=8+8 H=6+6 b=8 h=6
Once you've found the face area of the prism the length is 50.
That's a nice one.
But after you found the b1 and b2 I would use trigonometry instead.
The angle of the top right corner of the rectangle is the same that the angle of the trapisoid right bottom angle.
So, first the rectangle's angle: arctan(24/18) = 53,13°
Now the h: h = 6 * sin(53,13) => 6 * 0,8 = 4,8
From now everything the same.
I think this is a litle bit simplier.
People who mentally solved it while looking at the thumbnail
👇
I always make it to the end, and if there one thing i learned from this channel, it's that there is always a right angled triangle hidden somewhere
Good video!
I wonder if my geometry students could watch this once as a class and then reproduce what you did. Nothing alone would be new to them, but you do move a bit quicker than they might be able to follow.
Blew my mind with the triangle rotation
Wow was not expecting the flipping excitement. I made it to the end.
I made into the end!! Anyway yea you should do a video explaning the midpoints of a triangle or just straight up math!!❤❤
4 minutes and 20 seconds, nice.
What in the hell is the easy way to find the area of a trapezoid without knowing the height or base lengths lol
Find the area of the triangular prism that has the trapezoid, then subtract the triangular prism in the very corner.
@@BoolianKazooka YOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO I DIDJT EVEN SEE THAT
For the height of the trapezoid, u also coulda used the GMT theorems of similarity, immeadiately getting (h/6) = (8/10) as a proportion
No need for Pythagoras, it's a 3:4:5 triangle, and both triangles are similar with scale factor of 2.. I love the triangle area equivalence though, excellent trick for finding the height.
Nice, I love problems involving prismoidal volume which is applicable in earthworks (excavation and backfilling of soil).
Can you know for sure that the top and bottom of the trapezoid are parallel? The clue just says "prism", but there's nothing in the original diagram to assert that the lines in the base of the shaded area are actually parallel. I don't know if the word "prism" guarantees that does it? e.g. I don't know whether you can just "move the lower labels to the top" (1:06) can you?
I want the video about mid segmentes in triangles
Of course, we made to 4:20 minutes. I have never left one of your vids early. great content, definately helping me knock the cobwebs loose.
Never thought about the flipping triangle
Break up the base into 18 triangles of equal area. The shaded part is 3 out of 18, or 1/6 the base area. So the volume is (1/6)(18)(24)(50) which you can do in your head
I thought he was going to mention the fact that the triangle form a 3-4-5 special right triangle instead of working out Pythagorean Theorem.
thanks andy xoxo
Ah, relativity! 4 minutes isn't "long" for YT; although it's a bit long for you. Love your work!
1:06 isn't that an assumption? It's possible for the top 3 to not be divided into 6/6/6 like the bottom. There's nothing stopping the top segments from varying because they're not connected to the measurements at the bottom. The most I would assume is that the total length of the top is 18m because of the rectangular cross section 🤔
I made it to the end, after I’d worked it out in my head based on subtracting one 3:4:5 triangle from a bigger one.
Bro you must make A lot more of these videos
Wow, that is a massive prism
Great video man. Thanks
nice numbers, did it all in my head
Exciting, as usual
What a great video 😊 wow 🤩
4:20 length video again😂
Just teselate the larger triangle into 9 smaller similar triangle, 1-3-5, so 3 times the smallest triangle --> 24*3*50
Very nice - made it to the end for sure. How come you chose to do all the math for Pythagorean theorem when you could have pointed out that both triangles followed the 3-4-5 rule?
Video being 4 minutes and 20 seconds is absolutely not a mistake
The hard way indeed😂
I could never
hi andy. i made it to the end of the video. great visual effects.
Andy Math, please solve a OBMEP ( The Brazilian Mathematical Olympiad of Public Schools ) question to be more specific the 6th question of 2023 level 3 phase 1
He did it the HARD way!
No views in 20 seconds...
People aren't getting excited over math 😢
I made it to the end, waiting for you to abandon that method and just deal with the two 3 4 5 triangles on the end face ...
Guys he is getting stronger with each view, he is now able to interact with the math himself. :0
I was not prepared for the flip.
You did it but it was too lengthy you could also just subtract those triangles (the bigger triangle including shaded part and the small triangle)
i did it to the end
I just subtracted the area of the bigger prism with area of smaller prism. Calculation of hypotenuse is easy... In just 1 min I got 3600
How did we assume a rectangle with right angles, and parallel lines to make a trapezoid? (The spacing is not given for the top).
“This is probably gonna be a long video”
Me looking at the vid length: nice
I can spot another way, but it doesn't really seem easier. Let's have it.
I really need mid segments its so confusing
I got this one right with a bit of help🥳
Nice video length
And the easy way:
Face = triangle + parallellogram = 8*6/2 + 8*6 = 72
Volume = Face * Depth = 72 * 50 == 3600
I will assume that the shaded Area is 1/6 of whole.
As we have equal fragments.
One is smaler 2nd is average 3rd is the longest. So by averaging, we should have 1/6 of whole intersection field. And just multiply by length
Xd, i was right, WHO would gues
The easy way to do this is:
(16*12/2 - 8*6/2)*50 = 3600
(area of Triangle with shaded part - area of smaller unshaded triangle) * length
Simplifying all the way to 4.8 was an unnecessary step. You had 24/5. In the formula you were dividing by 2 and multiplying by 5. Those 10s would cancel out.
It's not always worth it to get all the way to decimal.
WAIT, THE BASE×HEIGHT THING WORKS IN EVERY ORIENTATION??????
S1=(3/18)*S
S=24*18
S1=3*24=72
V1=72*50=3600
Hey, I solved one! I did it the easy way, but still, I'm taking the win.
I want to do maths like u
What are the topics and sub topics I need to learn
Assume I know nothing and I'm complete beginner
I want to be like u guide me please
I watched to the end 🙂
3-4-5 triangles!
Nice 🙂
Made it to the end!
Is there eaven an easier way?
Made it to the end.
midsegment of triangle video!!!
I made it to the end 😊
I made it to the end ;)
How extending!
I made it to the end.
Actually it's called a cuboid
I made it to the end lol
i made it! it was fun
What is the easy way if you solved it the difficult way?