Polygons—FormulasGeometry CP1 (Holt 6-1) K.Santos

Polygon Interior Angle Sum

The sum of the measures of the angles of an n-gon is (n-2)180.

Example: 16-gon(n-2)180 with n=16(16-2)180(14)1802520

Why does this formula work?

1 triangle 2 triangles 3 triangles 180 360 540

1(180) 2(180) 3(180)(3-2)180 (4-2)180 (5-2)180 first number in the formula is number of sides

(n-2)180

Example—Finding a missing angleFind the missing angle measures.

First you have to find out what 121 112 the interior angle sum is. (n-2)180 78 (5-2)180 3(180) x x+3 540

Then write an equation to solve for the angles:121+112+78+x+x+3= 540314 +2x = 5402x = 226x = 113

So the angles measure: 113and 116 (x and x + 3)

Polygon—Exterior Angle Sum

The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

m<1+m<2+m<3+m<4+m<5 =360 1 5 2 3 4

Polygon—Exterior Angle Sum

For any polygon the exterior angle sum is 360

So, if it is a 50-gon then it’s exterior angle sum is 360

So, if it is a dodecagon then it’s exterior angle sum is 360.

Regular Polygon—One/each interior angle

To find one (or each) interior angle in a regular polygon with n, number of sides:

Example—Each interior angle in a regular polygon

Find the measure of each interior angle of a regular 18-gon.

each angle measures: 160

Regular Polygon—One/each exterior angle

To find one (or each) exterior angle in a regular polygon with n, number of sides:

Example—Each exterior angle in a regular polygon

Find the measures of each exterior angle in a regular 24-gon.

each angle measures: 15

Regular Polygon—Interior and Exterior Angle Relationships

Find each interior and exterior angle in a regular 40-gon.

Interior Angle Exterior Angle 9 exterior angle

171 interior angle

Notice that the interior angle and exterior angle are supplementary(add to 180. Exterior angle interior angle

Example—given the interior angle sum

Find the number of sides of a polygon that has an interior angle sum of 3960.

Interior angle sum formula: (n-2)180But we know what the sum is: 3960.So, (n-2)180 = 3960 180n -360=3960 180n = 4320

n = 24

The polygon has 24 sides (24-gon).

Example—given exterior angle sum

Find the number of sides of a polygon that has an exterior angle sum of 360.

Exterior angle sum of any polygon is 360.So, there is no way to find out how many sides the polygon has.

Example—regular polygon given one exterior angle

Find the number of sides of a regular polygon with an exterior angle measurement of 20.

The formula to find one exterior angle of a regular polygon is: But we know that the exterior angle is 20.

= 2020n = 360n = 18

So, the polygon has 18 sides (18-gon).

Example (1st method)—regular polygon given one interior angle

Find the number of sides of the regular polygon with one interior angle of 156.

The formula to find one interior angle of a regular polygon is: But we know the interior angle is: 156

= 156156n = (n-2)180156n = 180n – 360-24n = -360 n = 15

So, the polygon has 15 sides (15-gon).

But there is an easier way to do this problem…

Example (2nd method)—regular polygon given one interior angle

Find the number of sides of the regular polygon with one interior angle of 156.

If the interior angle is 156 we can find the exterior angle. 180-156 = 24.

So, now we know the exterior angle 24 and we know the formula to find one exterior angle:

= 2424n = 360 n = 15

So, the polygon has 15 sides (15-gon)

Total number of diagonals

To find the total number of diagonals: where n is the number of sides

Example: find the total number of diagonals in a 20-gon

170 diagonals