Polygons

MATH 102Contemporary Math

S. Rook

Overview

• Section 10.2 in the textbook:– Polygons– Interior angles of polygons– Similar polygons

Polygons

Polygons

• Consider drawing a figure on a piece of paper:– The figure is simple if it can be drawn without lifting the

pencil– The figure is closed if the starting point and ending point are

the same

• Polygon: a simple, closed figure, consisting of edges that are line segments (no curves) and at least three vertices– A polygon is regular if all its edges have the same length and

all its angles have the same measure– Polygons are named based on their number of edges (see

table on page 461)

Triangles

• A triangle is a three-sided polygon; the sum of its interior angles measures 180°

• Triangles can be classified further based on the lengths of their sides:– Scalene: no sides are equal & none

of the angles has the same measure– Isoceles: two sides have the same

length and the angles opposite them are equal in measure

– Equilateral: all three sides have the same length and the three angles have the same measure

Quadrilaterals

• A quadrilateral is a four-sided figure; the sum of its interior angles measure 360°

• Common quadrilaterals to recognize:– Trapezoid: at least one

pair of parallel sides– Parallelogram: a trapezoid,

but with two pairs of parallel sides• Each pair of sides has the same length

Quadrilaterals (Continued)

– Rectangle: a parallelogram, but with all its angles measuring 90°

– Rhombus: a parallelogram, but with the length of all its sides equal

– Square: a combination of a parallelogram and a rhombus

Interior Angles of Polygons

Sum of the Interior Angles of a Polygon

• We have already discussed the sum of interior angles for triangles and quadrilaterals

• There exists a relationship between the number of sides n of a polygon and the sum of its interior angles:– Sum of interior angles = (n – 2) x 180°– Theory can be found in Examples 1 & 2 on page 463

• Essentially the interior of any n-sided polygon can be broken up into n – 2 triangles

Interior Angles of a Polygon (Example)

Ex 1:

a) What is the sum of the interior angles of a dodecahdedron (12-sided polygon)?

b) If the dodecahderon in a) is regular, what is the measure of one of its interior angles?

c) How many sides does a polygon with the sum of its interior angles equaling 900° have?

Similar Polygons

Similar Polygons

• Two polygons are similar if their corresponding sides are proportional AND their corresponding angles have the same measure– Two triangles are similar if they have one pair of

corresponding sides and two equal angles

• Given that two polygons are similar, we can use a proportion with the corresponding sides to find the length of an unknown side– An expensive and time-consuming project is not usually

attempted before creating an inexpensive and scaled-down similar model!

Similar Polygons (Example)

Ex 2: Suppose we have drawn the following trapezoid:

Now suppose we decide to draw a second trapezoid similar to the first. If we extend the longer base to 10 inches, what would be the lengths of the remaining sides of the second trapezoid?

Similar Polygons (Example)

Ex 3:

Similar Polygons (Example)

Ex 4:

Summary

• After studying these slides, you should know how to do the following:– Define polygons and be able to name polygons according to their

number of sides– Understand the characteristics of special triangles and quadrilaterals– Find the sum of the interior angles for any polygon– Solve problems involving similar polygons

• Additional Practice:– See problems from Section 10.2

• Next Lesson:– Perimeter & Area (Section 10.3)