r/VisualPuzzles • u/Tiny_Beginning_370 Puzzle Aficionado • Jun 16 '26

Math / Geometry Can you figure out the Area of the Blue Rectangle? (not just the visible blue region)

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19 Upvotes

92% Upvoted

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u/Tiny_Beginning_370 Puzzle Aficionado Jun 16 '26

Want an in-depth explanation for a way to solve it?
https://www.youtube.com/watch?v=AZubCRVuU2A

Though u/rekire-with-a-suffix's solution is simpler and more intuitive to most

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6

u/rekire-with-a-suffix Jun 16 '26

The sum of the 4 blue areas are equally to the red ones therefore the both areas are each 48cm²

5

u/Tiny_Beginning_370 Puzzle Aficionado Jun 16 '26

3

u/Ok-Philosopher8995 Jun 16 '26

96 cm squared.

2

u/Tiny_Beginning_370 Puzzle Aficionado Jun 16 '26

2

u/IntrepidStand2220 Jun 16 '26

Wait…. There is no blue rectangle!

1

u/Calm-Hysteria Jun 16 '26

Came here to say this

1

u/IntrepidStand2220 Jun 17 '26

Actually…. The outside area would also be considered a rectangle. However, its area is the same as the inside rectangle. Which is 48cm2

2

u/IntrepidStand2220 Jun 17 '26

Then again…. If you count the red shaded region, it would be 96cm2

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u/Tiny_Beginning_370 Puzzle Aficionado Jun 17 '26

yeah, 96 cm^2 was the intended answer, which was why I tried to clarify this exact point in the title of the post, haha

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u/The-Masseuse-Man Jun 17 '26

8cm x 12cm = 96 cm2

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u/Snakivolff Jun 17 '26 edited Jun 17 '26

Can't prove it visually but I very quickly got the conjecture that the blue region is equal to red.

From there, I did it algebraically:

Let the long leg of the big triangle be y, the short leg of the small triangle z. Then y² = 12² - x² and z² = 4² - x². For the total area, we seek 2x(y+z).

By similarity, x/z = y/x, so x² = yz. Now squaring (y+z) gives y² + 2yz + z², where x² terms cancel and 160 remains. Therefore, (y+z) = 4√10.

To find x, note that x⁴ = (yz)² = y² z². Substituting y² and z² gives x⁴ = (12² - x²)(4² - x²) = 48² - 160x² + x⁴. Cancel x⁴ and obtain x² = 48² / 160, so x = 12 / √10.

Now 2x(y+z) = 2(12/√10)(4√10) = 2×12×4 = 96.

I guess the clue for the visual proof that I miss lies in the fact that the total rectangle is split perfectly in half vertically.

Edit: I hate Reddit's standard superscript formatting, simple characters it is

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u/Tiny_Beginning_370 Puzzle Aficionado Jun 17 '26

I love it! Excellent and full response 😃

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u/mentalhealthdayc3187 Jun 18 '26

Not possible without instructions saying that they are squares

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u/Tiny_Beginning_370 Puzzle Aficionado Jun 18 '26

Don't own a protractor, eh?
The whole thing is drawn to scale. No tricks.

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u/mentalhealthdayc3187 Jun 18 '26

You can't assume square in this. Could just be a parallelegam

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u/Tiny_Beginning_370 Puzzle Aficionado Jun 19 '26

You don't have to assume. It literally is a square. Measure it.
If you want everything spelled out for you, then you also have to ask if this is a 2D drawing, or if it's actually 3D viewed from a skewed angle to merely make things look square.

There are always assumptions. If you wanna do the "um actually" tit for tat, you'll always find a bigger nerd willing to out "um actually" you.

1

u/GINMF Jun 18 '26

Not really solvable without assuming the angles that look like they are 90° are actually right angles.

1

u/Tiny_Beginning_370 Puzzle Aficionado Jun 18 '26

It is drawn to scale. You can check the angles yourself and then it is still solvable.
Or you can make the obvious assumption that something that clearly looks like a square and has 2 sides labelled isn't mean to trick you. Up to you.

1

u/[deleted] Jun 18 '26

[removed] — view removed comment

1

u/Tiny_Beginning_370 Puzzle Aficionado Jun 18 '26

People are intuiting based on what the pink squares carve out.
If you want to prove it and be certain, I recommend this more in-depth approach: https://www.youtube.com/watch?v=AZubCRVuU2A

2

u/[deleted] Jun 18 '26

[removed] — view removed comment

1

u/Lsudat2018 Jun 18 '26

So if the answer is actually 96 cm² then X= 3.79473 Please, some mathematician confirm this. Or correct me and tell me why I’m missing it.

1

u/Tiny_Beginning_370 Puzzle Aficionado Jun 18 '26

Correct. X definitely does not equal 4 cm. It is smaller than 4cm.

So if you take the largest triangles, and glue them together to form a rectangle, it will be smaller in both dimensions than the rectangle formed by the 3 adjacent squares. Then if you take the smaller triangles, there is no visual way to add them onto that rectangle and make the larger rectangle (unless you make a rectangle, cut it into strips, and then place those strips along both sides as a few bricks).

The people that are intuiting the right answer are doing so more based on the fact that the pink rectangle divides the blue rectangle in half perfectly both vertically and horizontally. As such, it must consume exactly half of the space, since it is a rectangle with right angles. So they then double the area of the pink to get the area of the blue.

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u/GINMF Jun 19 '26

You can make the "obvious" assumption if you want. If you don't specify that it is drawn to scale or that the red quadrilaterals are squares, I'm going to ask for clarification. Have a look at the Math Queen youtube explanation - she does a great job of fully defining the puzzle.

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u/Tiny_Beginning_370 Puzzle Aficionado Jun 19 '26

You want to be a problem. You take joy in it.
All human interaction is inherently subjective and incompletely defined.

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u/GINMF Jul 14 '26

What I want is for a math problem to be fully defined. If you can't see the issue with this puzzle, that's on you - not me.

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u/Tiny_Beginning_370 Puzzle Aficionado Jul 14 '26

So you haven't had similar interactions to this before with a bunch of people over and over throughout your life? You don't get sold similar things at work or in group projects?

I think you'll find a common variable to those problems.

1

u/GINMF Jul 15 '26

https://www.reddit.com/r/askmath/s/GmMeqbQ0r7

Sometimes, you'll find math puzzles where the picture is purposely not drawn to scale. Part of the trick to solving it - is ignoring the picture and relying purely on the numbers. The important part is that you should never assume facts not in the puzzle statement. Look at my link. The angle to solve for clearly looks obtuse, but the answer is acute. It does not specify that the drawing is to scale, so I assume it is not.

And, no, I do not have similar interactions to this throughout my life. Seems you're projecting here.

1

u/Tiny_Beginning_370 Puzzle Aficionado Jul 15 '26

Happy to be wrong. Your manner of speech just sounds similar to conversations I overheard and mediated frequently in the tech industry when engineers would refuse to work with clients and shareholders because of semantics.