How to find the length of the diagonal of a hexagon - ACT Math
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The perimeter of a regular hexagon is 
. What is the length of one of its diagonals?
The perimeter of a regular hexagon is
To begin, calculate the side length of the hexagon. Since it is regular, its sides are of equal length. This means that a given side is 
or 
in length. Now, consider your figure like this:
The little triangle at the top forms an equilateral triangle. This means that all of its sides are 
. You could form six of these triangles in your figure if you desired. This means that the long diagonal is really just 
or 
.
To begin, calculate the side length of the hexagon. Since it is regular, its sides are of equal length. This means that a given side is
The little triangle at the top forms an equilateral triangle. This means that all of its sides are
Compare your answer with the correct one above
The figure above is a regular hexagon. O is the center of the figure. The line segment makes a perpendicular angle with the external side.
What is the length of the diagonal of the regular hexagon pictured above?
The figure above is a regular hexagon. O is the center of the figure. The line segment makes a perpendicular angle with the external side.
What is the length of the diagonal of the regular hexagon pictured above?
You could redraw your figure as follows. Notice that this kind of figure makes an equilateral triangle within the hexagon. This allows you to create a useful 
triangle.
The 
in the figure corresponds to 
in a reference 
triangle. The hypotenuse is 
in the reference triangle.
Therefore, we can say:
Solve for 
:
Rationalize the denominator:
Now, the diagonal of a regular hexagon is actually just double the length of this hypotenuse. (You could draw another equilateral triangle on the bottom and duplicate this same calculation set—if you wanted to spend extra time without need!) Thus, the length of the diagonal is:
You could redraw your figure as follows. Notice that this kind of figure makes an equilateral triangle within the hexagon. This allows you to create a useful
The
Therefore, we can say:
Solve for
Rationalize the denominator:
Now, the diagonal of a regular hexagon is actually just double the length of this hypotenuse. (You could draw another equilateral triangle on the bottom and duplicate this same calculation set—if you wanted to spend extra time without need!) Thus, the length of the diagonal is:
Compare your answer with the correct one above