Pentagon diagonals ...?

Question

Find the exact length of a regular pentagon with a side length 1

Answer 1

The diagonal is from one point, like the top, down to the lower right or lower left corner.


That has been given above: 1.618


If we make an isoceles triangle out of the top by drawing a line across the two adjacent vertices:

The top two sides are of equal length 1.
The inside angle (at the top) is 108°.
The other two angles are:
180° = 108° + θ + θ
or
2 • θ = 72°
or
θ = 36°


So, the distance across the bottom of that isoceles triange will be the length of the diagonal.

Well, half of the distance will be
½ • Diagonal = 1 • cos(36°)
or
the Diagonal = 2 • cos(36°)
or
the Diagonal ≈ 1.618


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but that is longer than the height, which is from the top straight down to the bottom, bisecting the bottom edge.

The height in this case is ≈ 1.539


Since the Diagonal ≈ 1.618, you can use the pythagorean theorem to solve for the height from the top vertex down to the mid-point of the bottom.

a² + b² = c²

c = the Diagonal ≈ 1.618
a = ½ (since the vertical line splits the bottom in half)

and solve for b, the height.

b = √( (1.618)² - (½)² )

or

the Height ≈ 1.539


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The End


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I Hope this Helps ❣ ❣

Answer 2

There are marvelous properties in the regular pentagon... it is full of the golden number... phi = (1 + sqrt(5))/2

One relation is this: the length of each diagonal by the length of the side is the golden number.

So diagonal / side = (1 + sqrt(5))/2

As the side = 1 ==> diagonal = (1 + sqrt(5))/2 = 1.618...

Answer 3

The diagonal is sqrt(1^2+1^2 -2*1*1cos(108))
= sqrt(2 +2cos72)

Answer 4

this is a famous problem and involves the golden ratio. the correct answer is

(1+SQRT(5) )/ 2

or about 1.618

Answer 5

unanswerable, as is