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Sum of Angles in Polygons
Introduction: A polygon is a closed plane (flat) figure with straight sides. There is a way to measure the sum of the interior angles that depends upon the number of sides (which is the same as the number of angles). We show this and also investigate the exterior angles.

The Lesson:
A polygon with the fewest number of sides is a triangle. A polygon in which all sides are equal in length and all angles have the same measure is called regular. We have shown that the three angles in a triangle have a sum of 180º. We use this fact to analyze polygons with more than three sides.

We show a quadrilateral and a pentagon below. One has four sides and angles, the other has five. Notice that by drawing diagonals within these polygons, we divide the polygons into triangles. The quadrilateral is composed of two triangles and the pentagon is composed of three.

Since the sum of the angles in a triangle is 180º, the sum of the angles in the quadrilateral is 360º because it is composed of two triangles. Similarly, we see that the sum of the five angles in the pentagon is 540º since it is composed of three triangles and 3 x 180º = 540º.

The number of triangles which compose the polygon is two less than the number of sides (angles). We generalize this result for a polygon of n sides and angles:
Theorem: The sum of the interior angles in a polygon with n sides is 180º(n – 2).
In the pentagon below, we have labeled the interior angles 1, 2, 3, 4, and 5. Each of these is supplementary respectively to exterior angles 6, 7, 8, 9, and 10. Therefore we have:
We know that angles 1 through 5 in a pentagon have a sum of 540º. We substitute 540º for these angles and we have: . Subtracting 540º from both sides, we can find the sum of the five exterior angles of this pentagon:
The sum of these exterior angles in any polygon will always be 360º, and although this is not a complete proof, we state the following:
Theorem: The sum of the exterior angles of a polygon is 360º.

Let's Practice:
  1. In a regular octagon (8 sides and angles), what is the measure of each interior angle and what is the measure of each exterior angle?
The sum of all 8 interior angles is given by .

Since each interior angle has the same measure, we divide by 8 to find the measure of one interior angle: .

Each exterior angle is supplementary to this and therefore must have measure 45º.
  1. The diagram given below shows a regular pentagon. If point O is the center, what are the measures of angles 1 and 2?
Since there are five sides in the pentagon, and each is of the same length, the measure of angle 2 must be .

The measure of angle 1 can be calculated from the triangle since this must be an isosceles triangle. The triangle must have 180º in addition to the 72º. Therefore angle 1 has a measure of half of 108º or 54º.

Another way to calculate the measure of angle 1 is to note that in a regular pentagon, each interior angle must have a measure of . Angle 1 is half of this, or 54º.

The angle at point O is called a central angle since it has a vertex at the center of the regular pentagon.

Example What is the measure of each interior and exterior angle in a regular decagon (10 sides and angles)?
What is your answer?
Example What is the measure of a central angle in a regular hexagon? ... in a regular octagon?
What is your answer?

M Ransom

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