Largest square (1 Viewer)

[jyu]

Find the area of the largest square that can fit inside a regular pentagon of unit side length.

 

[hscishard]

0.428 u^2?

no clue.

 

[jyu]

I have received another answer:
answer is area=(cot(pi/5))^2/2

I think the correct answer is greater than 1. The answer above is less than 1.

 

[hscishard]

Yea you get (cot36)^2 / 2 if you find the largest square inside the circle that fits in the pentagon.

it is obvious that there is a square that can have an area of 1 though...

 

[jyu]

This thread is getting cold.

 

[MrMMMan]

You want the answer?

 

[jyu]

your answer and explanation

 

[jyu]

Here is the answer



side = sin108 /sin63 = 1.067395682
area = 1.139333541

 

[MrMMMan]

I agree. At least 3 corners of the largest square must touch the sides of the pentagon or else a larger square can be fit by rotation/translation then dilatation. This leaves the square with all corners on pentagon's sides or the square you suggested, of which your one is the largest.

 

[jyu]

I agree. At least 3 corners of the largest square must touch the sides of the pentagon or else a larger square can be fit by rotation/translation then dilatation. This leaves the square with all corners on pentagon's sides or the square you suggested, of which your one is the largest.

Ok
Please draw a square with all corners on the regular pentagon and find its area. The largest and the smallest?

 

[MrMMMan]

This is the square:

As each angle of a regular pentagon is 108 degrees and for a square it is 90 degrees we can determine all angles in the figure. Hence the ratios between all line segments can also be determined. Given the side length of the pentagon is 1, the lengths of all other line segments is determined from these ratios.

I found: square length=sin 108/(sin 18+sin36)=1.0604… which is less than the length you have given.

What do you mean by largest and smallest?

 

[jyu]

This is the square: View attachment 22575

As each angle of a regular pentagon is 108 degrees and for a square it is 90 degrees we can determine all angles in the figure. Hence the ratios between all line segments can also be determined. Given the side length of the pentagon is 1, the lengths of all other line segments is determined from these ratios.

I found: square length=sin 108/(sin 18+sin36)=1.0604… which is less than the length you have given.

You've got it.

What do you mean by largest and smallest?

I wasn't sure whether you knew what you said.