# Proof that an N-gon is 180 Degrees

To prove that an n-gon is 180°, just divide the shape into triangles by drawing diagonals, then multiply 180° by the total number of triangles. **For example, a quadrilateral can be split into two triangles by drawing a diagonal from one corner to the opposite corner.** 180°x2=360°. Note that the number of triangles is always 2 less than the number of sides in the n-gon. For example, a pentagon will have three triangles and an octagon will have 6.

## Proof that an N-gon is 180 Degrees

Proof that an n-gon is 180 degrees. Let’s look at this chart. In the first column, we have different shapes. We’re going to fill in the rest (the number of sides, number of triangles, and the angle sum measure).

Let’s start with the triangle. **The triangle has three sides.** That’s only one triangle, right? We already know that the sum of the measures of the angles of a triangle is 180 degrees. Our next shape is a quadrilateral, or a rectangle in this case, but it would work with any quadrilateral. **Any four sided figure.**

If we draw one diagonal, then we split this rectangle up into two triangles. This quadrilateral has four sides and we split it up into two triangles. We know that each one of these triangles has an angle sum measure of 180 degrees. If you add 180 plus 180, or if you do 2 times 180, then you get 360 degrees.

**All quadrilaterals have an angle sum measure of 360 degrees.** Our next shape is a pentagon, which has five sides. When we draw our- we’ll put our five sides here first. When we draw our diagonals, we can split this pentagon up into 1, 2, 3 triangles this time. Again, each one of those triangles has an angle sum measure of 180 degrees.

Since we have three triangles, that’s 3 times 180, which gives us an angle sum measure of 540 degrees. The last shape we’re going to look at is a hexagon. This hexagon has six sides. Again, we’re going to split it up using our diagonals. When we draw our diagonals, we split our hexagon into 1, 2, 3, 4 triangles.

Again, each one of those triangles measures 180 degrees. When you multiply 180 times 4, you get 720 degrees. This should be enough information for us to see a pattern forming here. If we look at these two columns, our number of sides and our number of triangles, we notice that the number of triangles is always to less than our number of sides.

3 minus 2 is 1. 4 minus 2 is 2. 5 minus 2 is 3. 6 minus 2 is 4. If we used an n for our number of sides, we’re always subtracting two from that. Then, we take our number of triangles and we multiply it times 180, because each triangle is 180 degrees. We take our number of sides and we subtract 2 from that to find how many triangles our shape would be made up of.

Then, we multiply that times 180 degrees for the number of degrees in each one of our triangles. You can use this formula (n minus 2 times 180) to find the sum of the measures of the angles of any polygon.