[bushindo]

My math professor gave out this problem as a bonus during my first year of college. I bet the den will solve this in a jiffy.

Here is a picture of a 12-faces regular polyhedron. Each face is a regular pentagon, where all sides are equal and all interior angles are equal.

1) Find the angle between any two adjacent faces.

Bonus: Give the answer to number 1 in its exact form (ie. an expression containing square roots, trig functions, and whole numbers instead of giving the decimal form)

Edited June 5, 2009 by bushindo

[bonanova]

116.56...^o

I'll leave the bonus for another solver.

[Guest]

its 120

[Guest]

if you slice the figure it forms a six sided figure.

L

____

H / \ H

/ \

\ /

H \____/ H

L

Where L is the length of a side and H is the height of a pentagon. I do not know the height of a pentagon.

Oops you can't see my figure but since opposite sides are equal it is 120.

Edited June 5, 2009 by gkibarricade

[EventHorizon]

2 * arcsin( 1 / (2 sin (pi/5)))

Edited June 5, 2009 by EventHorizon

[bushindo]

2 * arcsin( 1 / (2 sin (pi/5)))

Congratulations to bonanova and EventHorizon for getting part I and the bonus, respectively. Well done.

[Guest]

OK I give up. Can some plase explain the method?

[bonanova]

OK I give up. Can some plase explain the method?

Consider the triangle formed by the midpoints of two adjacent

pentagonal faces and the midpoint of their common edge.

If you can determine these lengths, you can figure out the dihedral angle.

Google dihedral dodecahedron

[Guest]

Amarnath_Caterpillar

After long struggle 116.5 deg.

I did make a 3D model and measured it. Very tough excercise.

[Guest]

Thanks bonanova but I think I have done it anyway.

I did it by considering the tetrahedron formed by slicing through 3 ajacent faces.

116.5650512 degrees

My final calculation was:

Let x = sin(36)^2

Theta = arccos((2x-1)/2x))