Square Triangle
How many triangles have all three angles perfect squares (in degrees)?
Age
11 to 14
Challenge level
How many triangles have three angles that are all perfect squares, when measured in degrees?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
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 $10^2,8^2,4^2$