# Integer sides and angles in a right triangle.

#### MegaMoh

##### New member
As simple as the title says it. Is it possible to have a right triangle with 3 integer sides and 3 integer angles(30 degrees, 11 degrees, 79 degrees, etc.)

#### Subhotosh Khan

##### Super Moderator
Staff member
As simple as the title says it. Is it possible to have a right triangle with 3 integer sides and 3 integer angles(30 degrees, 11 degrees, 79 degrees, etc.)
I am assuming you want to measure angles in degrees - not in radians!

What will be the unit of measurements for the length of the sides - angstroms, cm, inches, light-years?

#### MarkFL

##### Super Moderator
Staff member
I would begin with a well-known method for generating Pythagorean triples:

$$\displaystyle (a,b,c)=(m^2-n^2,4mn,m^2+n^2)$$ where $$m,n\in\mathbb{N}$$ and $$n<m$$

And so, if you are using degrees to measure the angles, it will suffice to find an ordered pair $$(m,n)$$ such that:

$$\displaystyle \frac{180}{\pi}\arctan\left(\frac{m^2-n^2}{4mn}\right)=k$$ where $$k\in\mathbb{N}$$ and $$k<90$$.

I would likely use a computer to search for a solution.

#### MegaMoh

##### New member
I would begin with a well-known method for generating Pythagorean triples:

$$\displaystyle (a,b,c)=(m^2-n^2,4mn,m^2+n^2)$$ where $$m,n\in\mathbb{N}$$ and $$n<m$$

And so, if you are using degrees to measure the angles, it will suffice to find an ordered pair $$(m,n)$$ such that:

$$\displaystyle \frac{180}{\pi}\arctan\left(\frac{m^2-n^2}{4mn}\right)=k$$ where $$k\in\mathbb{N}$$ and $$k<90$$.

I would likely use a computer to search for a solution.
Shouldn't there be a condition so that $$\displaystyle m, n \neq 0$$?

#### MarkFL

##### Super Moderator
Staff member
Shouldn't there be a condition so that $$\displaystyle m, n \neq 0$$?
That is implied with $$m,n\in\mathbb{N}$$. The natural numbers are positive integers.

#### MegaMoh

##### New member
That is implied with $$m,n\in\mathbb{N}$$. The natural numbers are positive integers.
Oh, sorry. I thought it was $$\mathbb{R}$$ somehow

#### MegaMoh

##### New member
So isn't there some other fancy number theory way to get all solutions for this question without code?