How do you do this properly

[cara.mel]

This is a lenny conundrum from neopets xD
"In this diagram, the red part has an area of 800 square centimetres.



What is the area of the green part, in square centimetres? Please round up to the nearest whole number, and submit only the answer."

This is what I did:
Get paper and scientific calculator with good intentions, after 2 minutes got nowhere
Fetch graphics calculator
Randomly google and discover the thing is called a pentagram
Continue randomly googling and learn that all the lines in it are different by that golden ratio thing I've heard of before but don't understand beyond that
Google how to find the area of a pentagon
Find length of side -> now have all my sides of triangle
work it out from there

I am assuming there is a better method, ty ^_^
I also think it has something to do with a pentagon can be split into 3 isoceles triangles. meh

[Ahmad]

160 rt[5] 

[cara.mel]

How did you do that?

[Collin Li]

Using a few facts that can be taken off the internet:

* Sum of interior angles of a pentagon:
* Area of a regular pentagon (in terms of side length, S):

The geometric proofs of these claims are probably not incredibly difficult, but probably not that easy either, so it's easier just to start from here, and then clean up with simple geometry that we can do.

Interior angle of a pentagon:

Side length of the pentagon (using the area of 800 square centimetres):




The angle that appears twice in the isoceles triangle is:
(180 comes from the straight line that is made by a face of the regular pentagon and a face of a triangle - this also confirms the triangle is isoceles.)

Therefore, the angle at the far end is

Bisect the isoceles triangle to get a triangle with angle: and an opposite side length of:

Therefore, the adjacent side length is equal to:



The adjacent side length also happens to be the height of the triangle, so the area of the triangle is:




The fraction on the right simplifies to , confirming Ahmad's answer.

Simplification to come in the next post. Hey, I might as well research on how to derive those 2 facts I stated above (but I doubt I'll get that far).

[Ahmad]

How to find and hence using an algebraic method. (Can also be done using an intelligent geometric construction involving the golden ratio).

Note that:
has roots .
If we can find then we can take the real part or imaginary part to get or respectively.



Since, we want which is in the first quadrant on an argand diagram we can cancel off x=1.
(since )

Let
Then, we can write the above equation as, (Taking only positive).


Then you can use the tangent half angle formula:

[Ahmad]

* Sum of interior angles of a pentagon:
* Area of a regular pentagon (in terms of side length, S):

There's 5 triangles, with angles summing to 180 * 5 = 900. Now, the center angles add up to 360. So the sum of the interior angles of a regular pentagon is 900 - 360 = 540.

Drawing a perpendicular to the sides that meets with the center, immediately shows that as required.

[kingmar]

That's rape. Ahmad, I want your epsilons.

[/0]

Hi, someone posted a similar question on Mathhelpforum, the solution is quite nice:

http://www.mathhelpforum.com/math-help/geometry/24255-triangle-height.html

[Ahmad]

That's the same solution, using trigonometry. Coblin and I have simply elaborated (in simple steps) and added a few unnecessary extensions to get an answer without a calculator (exact form).