# What’s the big idea, 32.5 does not equal 31.5 [duplicate]

Here’s a visual proof for why 32.5=31.5

Here’s the animation

This puzzle came directly from Proofs without words on Mathoverflow.

Your Goal: Explain what’s wrong with the proof and where the “missing square” went! Good luck, this one drove me crazy for a bit!

• I'm almost positive this has been asked before but can't find a duplicate! (If it hasn't, I'm impressed it's lasted out so long - this is kind of a classic...)
– Stiv
Oct 29 '20 at 19:49
• @Stiv I looked too, I expected to find it but didn’t. Oct 29 '20 at 19:50
• @Stiv. Very similar concept in this very popular question puzzling.stackexchange.com/questions/24848/how-can-64-65/…
– DrD
Oct 29 '20 at 20:15
• @DrD does a similar concept mean a duplicate? Oct 29 '20 at 20:20
• In this case the answer logic is very similar to the 64=65 question. So it may not be a literal duplicate but close. Just my opinion.
– DrD
Oct 29 '20 at 20:23

## 2 Answers

The red and blue triangles don't have the same 'slope'.

So while in the first picture there is a 'dent' in the hypotenuse of the big triangle, in the second one there is a 'bump'.

That difference equates for the missing square.

Changing a bit the sizes of the pieces involved may help with visualising the illusion here. (sorry about the broken grid)

• Aw man, you beat me by like 50 seconds Oct 29 '20 at 19:48
• Nice clip, it has different triangles but it exaggerates the differences in angles better Oct 29 '20 at 21:10

It's not technically a triangle because the red triangle's hypotenuse has a different slope compared to the blue triangle's.

Rise over run for blue = 2/5

Rise over run for red = 3/8

These values would be the same if this were a triangle. Therefore, when the shapes are rearranged, the area changes.