I tried to derive the formula for the area of a pentagon using the golden ratio (not sine cosine or tangent), but my formula produces the wrong answer. I can't figure out where I've gone wrong.
I used the golden ratio to get the length of the diagonal.
Then, I used two diagonals two split the pentagon into 3 triangles: two identical 36-36-108, and one 36-72-72.
Using the pythagorean theorum, I calculated the height of each triangle and found the areas.
Last, I added my partial areas to get a total area and simplified.
s is the side length
A = s2*(Ø * sqrt(1-(Ø/2)2) + sqrt(Ø+.75)/2)
+118658 I'd be a lot happier if I was talking to a member. At least then I would have good reason to think you would see my answer !
You are not trying to find the area of any pentagon. You are trying to find that area of a regular pentagon!
Anyway, how can you get the height of the triangles without using trig to find the length of the diagonal first ?
Thank you for taking the time to respond. I didn't make an account as math isnt really my hobby (I prefer baduk!). This problem is just bothering me.
Since the angles of the triangle formed by making two diagonals from one point are 36-72-72, it is a golden triangle. So the ratio between the side of the pentagon and diagonal should be s : s*Ø, where Ø is the golden ratio: (1+sqrt(5))/2
I'm certain there's an error in my algebra somewhere.
The attached image shows my steps. a=s, φ = golden ratio. I checked it for sidelength four and my answer is off by what seems like a significant amount (27.7271... with my formula, 27.5276... for the normal formula). Since Ø2 = 1+Ø, Ø2-.25 = Ø+.75
Actually, chalk it up to calculator imput error ^^.
I simplified (using properties of phi) and got (s2/4)*(sqrt(5+2sqrt(5))+(sqrt(10+2sqrt(5)))). Tested with sides of four this produces the same result as the normal formula.
Which presents a different question:
Without using trigonometry, how would one simplify the above function to the normal function. For reference, here is standard area function: (s2/4)*sqrt(5(5+2sqrt(5)))
