How to show angles in a n-sided polygon add to 180° × (n-2)

To play this video you need to enable JavaScript in your browser.

This video can not be played

A short video showing how to prove the sum of the angles in a n-sided polygon is 180° × (n-2). For example a hexagon has 6 sides, so (n-2) is 4, and the internal angles add up to 180° × 4 = 720°.

How to show angles in a n-sided polygon add to 180° × (n-2)

Skip image gallery

Image caption,
Click to see a step-by-step slideshow.
Image caption,
WHAT YOU NEED: ruler, scissors, pencil, card/paper, coloured pens.
Image caption,
STEP 1: Use a ruler and pencil to draw an irregular hexagon. A hexagon has six sides so n=6. Use the scissors to cut the shape out.
Image caption,
STEP 2: Use a pen to draw on and colour in the angles. Then use a different pen and a ruler to draw a coloured line along each side.
Image caption,
STEP 3: Use a ruler to draw lines from the bottom left corner to the three opposite angles. We can ignore the two angles that are directly next to the starting point, because they already have lines along the sides linking them.
Image caption,
STEP 4: Use the scissors to cut along the lines. You should be left with four triangles. Notice that the number of triangles is two less than the number of sides of the original polygon. The angles in each triangle add up to 180°.
Image caption,
All the angles coloured in the original polygon are now in the triangles. To find the total we just multiply the number of triangles by 180°. So the sum of the angles in the original hexagon is 180° x (n-2) = 180° x 4 = 720°. You've done it!

How to show angles in a n-sided polygon add to 180° × (n-2)

More on Shape, space and measures