Angles and Diagonals in a Polygon ...........................

A polygon is a flat closed figure made from straight sides. In order to be a polygon, a figure needs to have at least three sides, but it can have more than that. Here are some polygons:

Pictured on the left are:
- the triangle, with three sides
- the quadrilateral, with four sides
- the pentagon, with five sides

If the sides are different lengths, then the shape of a polygon will not always be the same. There are many different triangles, for example, that are very different-looking, because their three sides are different from triangle to triangle.

If however the sides of the polygon are all the same length, then the polygon is called regular. Here are two examples.
A regular triangle has three equal sides. It is sometimes called an equilateral triangle. It could also be called equiangular, because its three angles are equal too.
A regular quadrilateral has four sides equal and four angles equal. It is also called a square.

But not all polygons are regular.

Here's a hexagon. Its sides are different lengths, so it's not the familiar regular hexagon shape of the honeycomb cell. But it's still a hexagon because it has six sides and six interior angles.

If you add up all six of the interior angles, what will the sum be?
In order to answer questions like this, you need to know how many degrees there are in the three angles of a single triangle. This is a fact you should know already ... three angles in a triangle add to 180 degrees.

Let's look at some polygons ...

First, a quadrilateral, with 4 sides. Pick one corner, and divide the figure into triangles as shown.

There are two triangles. Since each one has angles which add to 180 degrees, the total of all the angles in the quadrilateral must be 2 x 180 = 360 degrees.

Next a pentagon, with 5 sides. From one corner, you can split it into three triangles.

This makes the total of the angles inside a pentagon equal to 3 x 180 = 540 degrees.

Now the hexagon, with 6 sides. From one corner, you get four triangles.

This makes the total angles inside a hexagon equal to 4 x 180 = 720 degrees.

Do you see the pattern? It's the pattern that's important. It will let you work out the total degrees inside any polygon, even one with many more sides, where you might have difficulty drawing all the lines. Besides, that takes too long anyway!

You should have noticed that the number of triangles is always two less than the number of sides. And since there are always 180 degrees in a triangle, you will always be multiplying 180 by two less than the number of sides, to get the total interior angle.

As a formula, it looks like this:
So, for example, an eleven-sided polygon, which has 11 angles, should have a total angle inside, if you add up all the angles, of (11 - 2) x 180 = 9 x 180 = 1620 degrees

Now suppose the polygon is regular. That means all the angles are the same.
So if you know the total, you can also figure out what each angle is.

You already know how to do this for a regular triangle. Since its total interior angle is 180 degrees, and it has three equal angles, each one must be 60 degrees. That's why an equilateral triangle has three 60 degree angles. You knew that already.

But now you can do this for any regular polygon!

Here's a regular hexagon with 6 sides. The total degrees inside must be:
     (6 - 2) x 180
     = 4 x 180
     = 720
Since all the angles are the same, each one must be
     720 divided by 6 = 120 degrees.

Here's the formula you can use for any polygon:

So, for example, our 11-sided polygon, if it were regular, would have 11 angles, each one being

degrees

Finally, let's look at the diagonals of a polygon. A diagonal is any line joining the corners of a polygon, not counting the sides. A quadrilateral has two diagonals:

Here are the diagonals shown for a hexagon; there are nine diagonals:

Although there is a pattern, it's a little harder to see. And it gets difficult to draw all the diagonals without missing any. What we really want is the formula anyway, so here it is:

An eight-sided figure, an octagon, should have:

O.K., now it's time for you to work out a few. Do the three questions below first. Then we have a special calculator for you to play with; one that gives you the facts about polygons, and will float above this page. It has the formulas built in.
Give it a try here Enter the number of sides, press 'solve', and it does the rest!
Use the calculator answer the following three questions. When you're done, place the cursor over the square to the right of each answer to check it.

Questions to Try:

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