Lesson 9

Polygons

Let’s investigate polygons and their areas.

9.1: Which One Doesn’t Belong: Bases and Heights

Which one doesn't belong?

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

 

9.2: What Are Polygons?

Here are five polygons:

5 polygons

Here are six figures that are not polygons:

 six figures that are not polygons:

 

  1. Circle the figures that are polygons.

    Ten figures labeled A--J.

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

9.3: Quadrilateral Strategies

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

Six quadrilaterals labeled A--F.

 



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

Trapezoid, bases labeled b and a, height labeled 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.

9.4: Pinwheel

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

A shaded polygon on a grid.

 

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 \(A, B, C, D\), and \(E\).

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

Polygon with 5 sides

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 with a rectangle; subtracting the areas of the triangles from the area of the rectangle gives us the area of the polygon. 

Glossary Entries

  • opposite vertex

    For each side of a triangle, there is one vertex that is not on that side. This is the opposite vertex.

    For example, point \(A\) is the opposite vertex to side \(BC\).

    triangle with points labeled A, B, C.
  • polygon

    A polygon is a closed, two-dimensional shape with straight sides that do not cross each other.

    Figure \(ABCDE\) is an example of a polygon.

    Polygon with 5 sides