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'Square Triangle' printed from https://nrich.maths.org/
Answer: one triangle, with angles $100^\circ, 64^\circ, 16^\circ$
We want to find angles $x^\circ$, $y^\circ$ and $z^\circ$ so that $$x^2+y^2+z^2=180.$$
We know that the largest angle must be smaller than $180^\circ$ and bigger than $180^\circ/3 = 60^\circ$. So the largest angle must be $169^\circ, 144^\circ, 121^\circ, 100^\circ, 81^\circ$ or $64^\circ$.
largest angle |
medium angle |
smallest angle |
all squares? |
$169$ |
$9$ |
$2$ |
no |
|
$4$ |
$7$ |
no |
$144$ |
$36$ |
$0$ |
not a triangle |
|
$25$ |
$11$ |
no |
|
$16$ |
$20$ |
no |
$121$ |
$49$ |
$10$ |
no |
|
$36$ |
$23$ |
no |
$100$ |
$64$ |
$16$ |
yes! |
|
$49$ |
$31$ |
no |
|
$36$ |
$44$ |
no |
$81$ |
$81$ |
$18$ |
no |
|
$64$ |
$35$ |
no |
|
$49$ |
$50$ |
no |
$64$ |
$64$ |
$52$ |
no |
|
$49$ |
$67$ |
no |
So there is exactly one triangle with all three angles perfect squares, viz a triangle with angles $10^2,8^2,4^2$