Easy Ways to Calculate Polygon angles

Back to angles again, and you thought that you'd heard the last of them!

Interior Angles

You know now that all the interior angles of a triangle add up to 180o. You can use this gem of knowledge to make working out interior angles in other polygons really easy.

The quadrilateral below has been split into 2 triangles with the red line. The curved bits in case you were wondering are supposed to represent the angles.

If we have 2 triangles making up this shape, the sum (total) of the interior angles = 2 x 180 = 360o.

The same trick applies to other polygons, for example:

Here the sum of the interior angles = 3 x 180 = 540o.

There is a pattern emerging here…look at the number of sides on the first picture and compare it to the number of triangles.

First Picture: No. of sides = 4, No. of triangles = 2

Second Picture: No. of sides = 5, No. of triangles = 3

A polygon with 6 sides has 4 triangles, one with 7 sides has 5 triangles.

We can create a formula from this which will quickly tell us the number of triangles which will be found in a polygon of any number of sides. Writing down the formula in words we say:

The Sum of a Polygon's interior angles = The number of sides minus 2 multiplied by 180.

Using the shorthand letter n to represent the number of sides we get the formula:

Sum of interior angles = (n - 2) x 180.

The brackets are there to make sure that you subtract 2 from n before multiplying by 180.

Don't try and remember the formula, you don't need to - just remember splitting polygons up into triangles and the formula will appear magically from your memory.

One last thing on interior angles. If you have a regular polygon (pentagon, hexagon etc.) all the interior angles will be the same, so to find one interior angle you just divide the sum of the interior angles (which you now know how to calculate) by the number of sides.

 

Exterior Angles

Not much to say about these only that the interior angle of a polygon and the exterior angle both add up to 180o. A straight line is 180o and to see this more clearly look at the picture below:

The interior angles are shown in red and the exterior angles are shown in blue (interiors are normally warm and exteriors cold). 

 …there is one more snippet of information which is useful to know and that is that the sum of all the exterior angles of any polygon = 360o.