Polygons A many sided figure

The cross section of a brilliant-cut diamond forms a pentagon 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 pentagon is a type of polygon. Prefixes are used to name different types of polygons.

Polygon – a closed plane figure formed by three or more segments 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 (classified) based on the number of sides.

Properties of Triangles 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 sum of the interior angles of any polygon comes from the number of triangles in a figure

First remember that the sum of the interior angles of a polygon is given by the formula 180(n-2). A polygon is called a REGULAR 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) 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 180°, 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°, and there are four angles, so the measure of the interior angles are 90°.

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 3 triangles by connecting the vertices. That’s 2 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 not a regular pentagon. 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. So n = 8. Using our formula, that gives us 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 set the formula = 720° n - 2 = 4 divide both sides by 180 n = 6 add 2 to both sides

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

Practice with Sum of Interior Angles The sum of the interior angles of a hexagon. 360° 540° 720°

How many degrees are there in the sum of the interior angles of a 9-sided polygon? a) 1080° b) 1260° c) 1620°

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

How many sides does a polygon have if the sum of its interior angles is 2160°? a) 14 b) 16 c) 18

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.

Applying Geometric Properties Algebra in Geometry Applying Geometric Properties

Algebra can be used to solve many problems in geometry Algebra can be used to solve many problems in geometry. Using variables and algebraic expressions to represent unknown measures makes solving many problems easier. Find the sum of interior angles using the formula. 180°(n - 2) = 180°(4 – 2) = 180°(2) = 360° Set the sum of the angles equal to the total. 120° + 50° + 80° + x = 360° 250° + x = 360° 250 – 250 + x = 360 -250 x = 110°

Remember, a regular polygon has congruent sides and congruent angles Remember, a regular polygon has congruent sides and congruent angles. Given the regular pentagon at the left, what are the measures of the interior angles. (use the formula) 180°(n – 2) = 180°(5 – 2) = 180°(3) = 540° # of angles = 5 540°/5 = 108° Each angle in a regular pentagon measures 108°

Using geometry to solve word problems. Remember, draw a picture. Quadrilateral STUV has angle measures of: (3x + 15)° (2x + 20)° (4x + 5)° (2x – 10)°, add the angles = 360 (3x + 15) + (2x + 20) + (4x + 5) + (2x – 10) = 360 11x + 30 = 360 11x = 330 x = 30° x = 30°, then 3x + 15 = 3(30) + 15 = 105° 2x + 20 = 2(30) + 20 = 80° 4x + 5 = 4(30) + 5 = 125° 2x – 10 = 2(30) – 10 = 50° So, 105° + 80° + 125° + 50° = 360°

Solve the following: Figure ABCDEF is a convex polygon with the following angle measures. What is the measure of each angle? (draw a picture) A = 4x B = 2x C = 3x D = 5x + 10 E = 3x – 20 F = 2x – 30 Answer »»

(4x) + (2x) + (3x) + (5x + 10) + (3x – 20) + (2x – 30) = 720° 19x – 40 = 720° 19x = 720° x = 40°, so 4x = 4(40) = 160° 2x = 2(40) = 80° 3x = 3(40) = 120° 5x + 10 = 5(40) + 10 = 210° 3x – 20 = 3(40) – 20 = 100° 2x – 30 = 2(40) – 30 = 50° check, 160° + 80° + 120° + 210° + 100° + 50° = 720° 720° = 720°

Polygons Problem Solving

1) Find the sum of the angle measures in the figure to the left 1) Find the sum of the angle measures in the figure to the left. a) 180° b) 540° c) 720° d) 1260°

2) Find the angle measures in the polygon to the right 2) Find the angle measures in the polygon to the right. a) m° = 150° b) m° = 144° c) m° = 120° d) m° = 90°

3) Give all the names that apply to the figure at the left 3) Give all the names that apply to the figure at the left. a) quadrilateral, square, rectangle, rhombus, parallelogram b) quadrilateral, trapezoid c) quadrilateral, parallelogram, rectangle, square d) quadrilateral, parallelogram, trapezoid

4) Find the sum of the angle measures in a 20-gon 4) Find the sum of the angle measures in a 20-gon. If the polygon is regular, find the measure of each angle. a) 198°, 9.9° b) 720°, 72° c) 1800°, 90° d) 3240°, 162°

5) Find the value of the variable 5) Find the value of the variable. a) x° = 90° b) x° = 110° c) x° = 120° d) x° = 290°

6) Given the polygon at the left, what is the measure of the interior angles? A) 720 B) 540 C) 360 D) 180