Assistance needed  How do we know that that angle is 60 degrees?

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[Originally posted in Discussions]
How do we know that that angle is 60 degrees? Did we just guess that the total is 360 degrees for the entire outside of the hexagon or something???
And 150 times the square root of 3 doesn't simplify into anything and we just leave it like that?
And sorry but I don't understand the formula at 14 minutes at all [ A = (S/2) squared x (multiplied by) the root of 3]
AAAAAAAAAAAAAand for this one how do we know that shortest side length is 1 when the hypotenuse is 2?

For the answer to why the short side has length 1, the reason is because this triangle is a 306090 right triangle, which is explained in this other post: https://daily.poshenloh.com/manage/discussion/posts/268087
The reason Prof. Loh takes 360 and divides by 6 is because if you just consider which angle the ant is pointing at, the ant makes one complete revolution as he goes around the hexagon. Pretend that you don't draw his straight paths, and instead pretend that he walks in place. He still turns, though. Do you see that he spins around one complete revolution, which is equal to 360 degrees?
The formula at 14 minutes relates again to the ratios of lengths of a 306090 right triangle, which is basically an equilateral triangle cut in half. The squares are applied because the Pythagorean Theorem is used.
Happy Learning!
The Daily Challenge Team

Is there some other way to simplify the answer or do you just leave it 150 times the square root of 3?

@TSSGraviser
There's no way to further simplify \( 150 \sqrt{3} \), so you would leave it as is.Happy Learning,
The Daily Challenge Team