Polygons

A many sided figure

The cross section of a brilliant-cut diamond forms a pentagon. The most beautiful and valuable diamonds have precisely cut angles that maximize the amount of light they reflect.

A penta gon is a type of poly gon .

Prefixes are used to name different types of polygons.

Polygon – a closed plane figure formed by three or more segments.

Regular polygon – a polygon with congruent sides and angles.

Prefixes used to name polygons: tri-, quad-, penta-, hexa-, hepta-, octa-, nona-, deca Polygons are named number of sides.

( classified ) based on the

Polygons

Properties of polygons, interior angles of polygons including triangles, quadrilaterals, pentagons, heptagons, octagons, nonagons, and decagons.

Properties of Triangles

Triangle – 3-sided polygon The sum of the angles in any triangle is 180 ° (triangle sum theorem)

The formula we use to find the interior angles of any sum of the polygon comes from the number of triangles in a figure

First remember that the sum of the REGULAR interior formula angles of a polygon is given by the 180(n-2).

A polygon is called a when all the sides are congruent and all the angles are congruent.

The picture shown to the left is that of a Regular Pentagon . We know that to find the sum of its interior angles we substitute n = 5 in the formula and get: 180(5 -2) = 180(3) = 540 °

Regular triangles - Equilateral

All sides are the same length ( congruent 180 ° ( ) and all interior angles are the same size congruent ).

To find the measure of the interior angles, we know that the sum of all the angles equal , and there are three angles.

So, the measure of the interior angles of an equilateral triangle is 60 ° .

Quadrilaterals – squares

All sides are the same length (congruent) and all interior angles are the same size (congruent) To find the measure of the interior angles, we know that the sum of the angles equal 360 are 90 ° .

° , and there are four angles, so the measure of the interior angles

Pentagon – a 5-sided polygon

To find the sum of the interior angles of a pentagon, we divide the pentagon into triangles. There are three triangles and because the sum of each triangle is 180 ° we get 540 ° , so the measure of the interior angles of a regular pentagon is 540 °

Hexagon – a 6-sided polygon

To find the sum of the interior angles of a hexagon we divide the hexagon into triangles. There are four triangles and because the sum of the angles in a triangle is 180 ° , we get 720 ° , so the measure of the interior angles of a regular hexagon is 720 ° .

Octagon – an 8-sided polygon

All sides are the same length (congruent) and all interior angles are the same size (congruent).

What is the sum of the angles in a regular octagon?

Nonagon – a 9-sided polygon

All sides are the same length (congruent) and all interior angles are the same size (congruent).

What is the sum of the interior angles of a regular nonagon?

Decagon – a 10-sided polygon

All sides are the same length (congruent) and all interior angles are the same size (congruent).

What is the sum of the interior angles of a regular decagon?

Using the pentagon example, we can come up with a formula that works for all polygons.

Notice that a pentagon has

5

sides, and that you can form triangles by connecting the vertices. That’s

2 3

less than the number of sides. If we represent the number of sides of a polygon as

n

, then the number of triangles you can form is

(n-2)

. Since each triangle contains

180

° , that gives us the formula: sum of interior angles =

180(n-2)

Warning !

• Look at the pentagon to the right. Do angle E and angle B look like they have the same measures? You’re right---they don’t. This pentagon is pentagon.

not a regular • If the angles of a polygon do not all have the same measure, then we can’t find the measure of any one of the angles just by knowing their sum .

Using the Formula

Example 1: Find the number of degrees in the sum of the interior angles of an octagon.

An octagon has 8 sides that gives us . So n = 8 . Using our formula, 180(8-2) = 180(6) = 1080 °

Example 2: How many sides does a polygon have if the sum of its interior angles is 720 ° ?

Since, this time, we know the number of degrees , we set the formula equal to 720 ° , and solve for n .

180(n-2) = 720 n - 2 = 4 set the formula = 720 divide both sides by 180 ° n = 6 add 2 to both sides

Names of Polygons

Triangle Quadrilateral Pentagon Hexagon Heptagon or Septagon Octagon Nonagon or Novagon Decagon 3 sides 4 sides 5 sides 6 sides 7 sides 8 sides 9 sides 10 sides

Practice with Sum of Interior Angles

1) The sum of the interior angles of a hexagon.

a) 360 ° b) 540 ° c) 720 °

2) How many degrees are there in the sum of the interior angles of a 9-sided polygon?

a) 1080 ° b) 1260 ° c) 1620 °

3) If the sum of the interior angles of a polygon equals 900 ° , how many sides does the polygon have?

a) 7 b) 9 c) 10

4) How many sides does a polygon have if the sum of its interior angles is 2160 ° ?

a) 14 b) 16 c) 18

5) What is the name of a polygon if the sum of its interior angles equals 1440 ° ?

a) octagon b) decagon c) pentagon

Special Quadrilaterals

4-sided figures

Quadrilaterals with certain properties are given additional names.

A square has 4 congruent sides and 4 right angles.

A rectangle has 4 right angles.

A parallelogram has 2 pairs of parallel sides.

A rhombus has 4 congruent sides.

A kite has 2 sets of adjacent sides that are the same length (congruent) and one set of opposite angles that are congruent.