ACT Math : How to find the length of the diagonal of a hexagon

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #31 : Geometry

The perimeter of a regular hexagon is \displaystyle 105. What is the length of one of its diagonals?

Possible Answers:

\displaystyle 17.5

\displaystyle 35

\displaystyle 17.5\sqrt{2}

\displaystyle 52.5

\displaystyle 26.25

Correct answer:

\displaystyle 35

Explanation:

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 \displaystyle \frac{105}{6} or \displaystyle 17.5 in length. Now, consider your figure like this: 

Hex175

The little triangle at the top forms an equilateral triangle. This means that all of its sides are \displaystyle 17.5. You could form six of these triangles in your figure if you desired. This means that the long diagonal is really just \displaystyle 2*17.5 or \displaystyle 35.

Example Question #31 : Act Math

Hexcenter71

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?

Possible Answers:

\displaystyle 14

\displaystyle 42

\displaystyle \frac{28\sqrt{3}}{3}

\displaystyle 14\sqrt{2}

\displaystyle 7\sqrt{2}

Correct answer:

\displaystyle \frac{28\sqrt{3}}{3}

Explanation:

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 \displaystyle 30-60-90 triangle.

Hexcenter72

The \displaystyle 7 in the figure corresponds to \displaystyle \sqrt{3} in a reference \displaystyle 30-60-90 triangle. The hypotenuse is \displaystyle 2 in the reference triangle. 

Therefore, we can say:

\displaystyle \frac{7}{\sqrt{3}}=\frac{h}{2}

Solve for \displaystyle h:

\displaystyle h=\frac{14}{\sqrt{3}}

Rationalize the denominator:

\displaystyle h=\frac{14}{\sqrt{3}}*\frac{\sqrt{3}}{\sqrt{3}}=\frac{14\sqrt{3}}{3}

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:

\displaystyle \frac{28\sqrt{3}}{3}

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