Drawing a Decagon
It is not easy to draw a regular decagon without tools.
On a piece of writing paper (with equally spaced lines), I am trying to draw a regular decagon, as shown above. I started by drawing two sides so that their vertical extent is precisely 1 unit of the paper (black lines).
Now I want to draw the next side (red line), and I wonder how far it will extend vertically. To 3 decimal places, what is the distance marked with a question mark?
The answer is 0.61803399.
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3 solutions
Isn't golden ratio one more than the answer?
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It depends on your definition. If you write the golden ratio as larger : smaller, then its value is 1.618. If you write it as smaller : larger, then its value is 0.618.
In fact, the golden ratio can be defined as the ratio which differs by 1 from its inverse. :)
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Golden ratio has 2 values, (1+√5)/2 and (1-√5)/2 from the equation which says x^2-1=x.
It not only differs by one it's actually one more than it's inverse.
The angle between the sides of a 10-gon = 180 - 360/10 =144.
Therefore between top black and horizontal the angle is 144/2=72 by symmetry of two black sides about the
horizontal line.
Vertical projection of the side is =1. So side=1/Sin72.
But then red side is turned by 360/10=36 degrees clock wise from the top black .
Implies red side is 72 - 36 = 36 with horizontal.
So vertical projection is
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.
Using simple geometry, one may get the required distance as
1 − 4 sin 2 1 8 ∘ 4 sin 2 1 8 ∘ ≈ 0 . 6 1 8 0 3 3 9 8 8 c m
Method 1 : Apply the apothem formula, apothem, a = r × cos n 1 8 0 ∘ , where r denote the circumradius of this poylgon and n denote the number of sides of this polygon, n = 1 0 .
By referring to the right triangle formed above, the angle formed between the hypothenuse ( r ), and the horizontal line is 1 0 3 6 0 ∘ = 3 6 ∘ and the height of this right triangle is just 1 cm, then sin 3 6 ∘ = r 1 ⇒ r = sin 3 6 ∘ 1 .
The apothem of this regular decagon is just r ⋅ cos ( 1 0 1 8 0 ∘ ) = sin 3 6 ∘ cos 1 8 ∘ = 2 cos 1 8 ∘ sin 1 8 ∘ cos 1 8 ∘ = 2 sin 1 8 ∘ 1 = 2 5 + 1 .
So the distance in question is just 2 5 + 1 − 1 = 2 5 − 1 ≈ 0 . 6 1 8 .
Method 2 : Without applying the apothem formula.
From one side to another in a regular decagon involves a turn of 3 6 0 / 1 0 = 3 6 ∘ .
The two black lines make equal angles with the vertical; this must be 3 6 / 2 = 1 8 ∘ . The red line will therefore make an angle of 1 8 + 3 6 = 5 4 ∘ with the vertical.
If the side of the polygon is a , the vertical extent of the black lines is y b = a cos 1 8 ∘ , and that of the red line is y r = a cos 5 4 ∘ . Therefore h = y b y r = a cos 1 8 ∘ a cos 5 4 ∘ = cos 1 8 ∘ sin 3 6 ∘ = 2 sin 1 8 ∘ = 2 ( 4 1 5 − 4 1 ) = 2 1 5 − 2 1 ≈ 0 . 6 1 8 . Incidentally, this is the golden ratio.