Lesson 9Polygons

Learning Goal

Let’s investigate polygons and their areas.

Learning Targets

  • I can describe the characteristics of a polygon using mathematical vocabulary.

  • I can reason about the area of any polygon by decomposing and rearranging it, and by using what I know about rectangles and triangles.

Lesson Terms

  • edge
  • polygon
  • quadrilateral
  • vertex

Warm Up: Which One Doesn’t Belong: Bases and Heights

Problem 1

Which one doesn’t belong?

Four triangles on a grid, labeled S, T, U, and V.

  1. Triangle

  2. Triangle

  3. Triangle

  4. Triangle

Activity 1: What Are Polygons?

Problem 1

Here are five polygons:

5 polygons with different shapes

Here are six figures that are not polygons:

6 figures that are not polygons

  1. Select the figures that are polygons.

    1. A square with a diagonal line drawn off the top left corner.
    2. A shape like a starburst.
    3. A cube
    4. A curved arrow
    5. A pentagon
    6. An irregular closed shape.
    7. A trapezoid
    8. A blue triangle
    9. An irregular shape composed of three unconnected triangles.
    10. A rectangle with 2 vertical lines drawn inside of it, each one near the side.

  2. What do the figures you circled have in common? What characteristics helped you decide whether a figure was a polygon?

Activity 2: Quadrilateral Strategies

Problem 1

Find the area of two quadrilaterals of your choice. Show your reasoning.

Six quadrilaterals labeled A--F.

Are you ready for more?

Problem 1

Here is a trapezoid. and represent the lengths of its bottom and top sides. The segment labeled represents its height; it is perpendicular to both the top and bottom sides.

A trapezoid labeled with sides of a and b and height of h.

Apply area-reasoning strategies—decomposing, rearranging, duplicating, etc.—on the trapezoid so that you have one or more shapes with areas that you already know how to find.

Use the shapes to help you write a formula for the area of a trapezoid. Show your reasoning.

Activity 3: Pinwheel

Problem 1

Find the area of the shaded region in square units. Show your reasoning.

A shaded polygon on a grid.

Lesson Summary

A polygon is a two-dimensional figure composed of straight line segments.

  • Each end of a line segment connects to one other line segment. The point where two segments connect is a vertex. The plural of vertex is vertices.

  • The segments are called the edges or sides of the polygon. The sides never cross each other. There are always an equal number of vertices and sides.

Here is a polygon with 5 sides. The vertices are labeled , and .

A polygon encloses a region. To find the area of a polygon is to find the area of the region inside it.

An enclosed polygon of an irregular shape labeled A, B, C, D, E.

We can find the area of a polygon by decomposing the region inside it into triangles and rectangles.

Three identical five-sided polygons. The first two are divided up into triangles in rectangles. The third is surrounded by a rectangle, the area of which outside the polygon is shaded.

The first two diagrams show the polygon decomposed into triangles and rectangles; the sum of their areas is the area of the polygon. The last diagram shows the polygon enclosed in a rectangle; subtracting the areas of the triangles from the area of the rectangle gives us the area of the polygon.