Study Guide

Basic Geometry - Angles in a Polygon

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Angles in a Polygon

As we discussed before, the three angles of a triangle always add up to 180°.

2 ABC Triangles

In each case m < A + m < B + m < C = 180 degrees. By the way, m < A means "the measurement of angle A."

To find the total number of degrees in any polygon, all we have to do is divide the shape into triangles. To do this start from any vertex and draw diagonals to all non-adjacent vertices.

Here is a quadrilateral.2 Quadrilaterals
If we draw all the diagonals from a vertex we get two triangles.
Each triangle has 180°, so 2 ×180° = 360° in a quadrilateral. 
Pentagon – 5 sidesPentagon3 triangles × 180° = 540°
Hexagon – 6 sidesHexagon4 triangles × 180° = 720°
Septagon – 7 sidesSeptagon5 triangles × 180° = 900°
Octagon – 8 sidesOctagon6 triangles × 180° = 1080°

Are you noticing a pattern? Turns out, the number of triangles formed by drawing the diagonals is two less than the number of sides. If we use the variable n to equal the number of sides, then we could find a formula to calculate the number of degrees in any polygon. This is the formula for the sum of the interior angles in a polygon with n sides:

Formula for the sum of the interior angles of an n-gon = (n-2) x 180 degrees

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