Is there a way to determine the angle measure of a regular polygon in hyperbolic space?

Ashlynn Hale

Ashlynn Hale

Answered question

2022-08-11

Hyperbolic Angle Measure of Polygons
Is there a way to determine the angle measure of a regular polygon in hyperbolic space? I know that this depends on the length of the sides. A an example, I know that for an equilateral triangle with side length a and angle A, then sec A = 1 + 2 e a e 2 a + 1 .
Is there a similar formula for higher regular polygons?

Answer & Explanation

Dwayne Hood

Dwayne Hood

Beginner2022-08-12Added 10 answers

Step 1
You can use the hyperbolic cosine rule for angles, namely cos α = cos β cos γ + sin β sin γ cosh a , on a triangle with vertices at the centre of the polygon and two consecutive vertices of the polygon. Taking a as the side length, the central angle is obviously 2 π / n and the triangle is isosceles, so β = γ, which gives
cos 2 π n = cos 2 β + sin 2 β cosh a = 1 2 ( 1 + cos 2 β ) + 1 2 ( 1 cos 2 β ) cosh a .
Step 2
The internal angle A is then 2 β, so we find after some hyperbolic identities that sec A = tanh 2 1 2 a cos 2 π n sech 2 1 2 a

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school geometry

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?