Math Problem
 

Last week I entered a math competition and one of their questions was. There is a hexagon with each side with length of 5 cm. What is the total length of all of its diagonal. I was really put off by this question, but then I thought that diagonals would cut each other and make 90 degree angles each.
Then I know one side is 5 cm, and its a 90 degree angle, so it'd be like 45 45 90 triangles, in which the ratio is 1, 1, square root of two respectively.
Doing then all the proportion and math, i got 15 multiplied by square root of two or approximately 21.213 Is there any math expert out there who can verify my answer, help is appreciated :)
Thank you. Hope it all makes sense.

not quite. for the diagonals that radiate from the center, they make angles of 60 degrees (360/6). there are other diagonals that don't go through the center. I believe there are a total of nine diagonals

Yeah, I see that now. Well I got that wrong.
So how would anyone solve that?

Split the hexagon into triangles, and then use pythagoras to work out the angles and distances.

I havent done trigonometry for over 3 years.. lets see if it stuck...

lengthOfDiagType1 = 5 / (sin 120) = 5.7735 (From sin 120 = 5 / Hypoteneuse)
lengthOfDiagType2 = 2 * sideLength = 2 * 5 = 10

count(diagType1) = 6
count(diagType2) = 3

(6 * lengthOfDiagType1) + (3 * lengthOfDiagType2) = 64.641 cm (to 3 dp)

Could someone check that please, its very possible that i got it wrong hehe.

No pythagoras here zen_tom :)

P.S. sorry i listed it like a program, thats just what i'm good at...

Practical solution:
Get 7 rulers.
Take 6 of the 7 rulers, construct a hexagon with side length 5cm.
Take the 7th ruler, measure the diagonals and add them up. :D

Thanks Welshbyte, hope I had done that.
FngKestrel, heh next time I'll ask them for a ruler.:)

I get a different answer than welshbyte. The "long" diagonals are indeed 10cm long, but for the shorter ones I get 5*sqrt(3) or ~8.66.

I get this answer from the Law on Sines: x/sin(120) = 5/sin(30).

The total length would be 3*10 + 6*(5*sqrt(3)), or ~81.96.