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A regular hexagon has a perimeter of $p$ (in length units) and an area of $A$ (in square units).  If $A = \frac{3}{2},$ then find the side length of the hexagon.

 Jan 7, 2024
 #1
avatar+129487 
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If we inscribe the hexagon in a circle, the radius of the  circle  = the side length of the hexagon

 

The hexagon is  composed of  6 equilateral triangles  each having an area   =

(1/2)r^2 sin (60°)  = (1/2)r^2 * sqrt (3) / 2

 

So

 

A = 6 (1/2) r^2  * sqrt (3)  / 2

 

3/2 =  3r^2 *sqrt (3) / 2

 

1 = r^2 sqrt (3)

 

1 / sqrt (3)  = r^2

 

r  =  1 / 3^(1/4)  ≈  .76  = side of the hexagon

 

 

cool cool cool

 Jan 7, 2024
 #1
avatar+129487 
+1

For a hexagon, the area = 6 (1/2) (side)^2 *sqrt (3) /2

 

3/2  = 6 (1/2)(side)^2 sqrt (3) / 2

 

3/2 = 3side^2 sqrt (3) /2

 

1 = side^2 *sqrt (3)

 

1/sqrt 3 = side ^2

 

side =  sqrt  [ 1 /sqrt 3 ] ≈ .76

 

cool cool cool

CPhill Jan 16, 2024

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