A hexagon is a polygon with six sides. In a regular hexagon all sides and angles are equal. In geometry, you might be given a problem where you know how tall or wide a regular hexagon is (for example, a given hexagon might measure 12cm from the middle of one side to the middle of another), and you are asked to find the length of one hexagon side. The problem becomes simpler when you realise that a regular hexagon can be divided into six equal-sized equilateral triangles, and so you can use a basic trigonometric identity to find the length of one side of such a triangle.

- Skill level:
- Challenging

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### Things you need

- Ruler
- Pencil
- Calculator with square root function

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## Instructions

- 1
Divide the hexagon into six equal triangles. Each edge of the hexagon should be the base of one of the triangles, and all the triangles should meet at a point in the centre. This helps you to visualise the problem, but you can skip this step if you are comfortable with the idea that a hexagon can be formed from six triangles.

- 2
Divide the given height of the hexagon by two. For example, if it is 12cm from the bottom side of a hexagon to the top side, divide 12 by 2. This gives you the height of one of the equilateral triangles, 6cm.

- 3
Use your result from Step 2 in the following formula to find the length, S, of one side. In the formula H is the height you found in Step 2.

S = sqrt [ ( 4*H^2 ) / 3 ]

Apply the formula as shown in Steps 4 to 6.

- 4
Square the height, H. In our example, 6cm squared is 36cm.

- 5
Multiply the result of Step 4 by 4 and divide by 3. 4*36/3 is 48cm.

- 6
Take the square root of Step 5. The square root of 48cm is 6.93cm.

The length of one side of the hexagon is 6.93cm.

#### Tips and warnings

- If you have a hexagon that's measured from one vertex (corner) across to the opposite vertex, you don't need to use a formula: just divide the width by 2, and that's your answer. For example, a hexagon that measures 10cm from one vertex to the opposite vertex has sides of 5cm.