Area of Polygons - Composition and Decomposition

Videos and solutions to help Grade 6 students find the area of polygons by decomposing into other triangles and polygons.

New York State Common Core Math Grade 6, Module 5, Lesson 5

Download lessons for 6th Grade

Lesson 5 Student Outcomes

Students show the area formula for the region bounded by a polygon by decomposing the region into triangles and other polygons. They understand that the area of a polygon is actually the area of the region bounded by the polygon.

Students find the area for the region bounded by a trapezoid by decomposing the region into two triangles. They understand that the area of a trapezoid is actually the area of the region bounded by the trapezoid. Students decompose rectangles to determine the area of other quadrilaterals.

Example 1: Decomposing Polygons into Rectangles
We need to decompose irregularly shaped polygons into rectangles. We need to make a choice of where to separate the figure. It may often involve calculating the length of unknown sides of the new figures.

The Intermediate School is producing a play that needs a special stage built. A diagram is shown below (not to scale).

a. On the first diagram, divide the stage into three rectangles using two horizontal lines. Find the dimensions of these rectangles and calculate the area of each. Then find the total area of the stage.

b. On the second diagram, divide the stage into three rectangles using two vertical lines. Find the dimensions of these rectangles and calculate the area of each. Then find the total area of the stage.

c. On the third diagram, divide the stage into three rectangles using one horizontal line and one vertical line. Find the dimensions of these rectangles and calculate the area of each. Then find the total area of the stage.

d. Think of this as a large rectangle with a piece removed.
i. What are the dimensions of the large rectangle and the small rectangle?
ii. What are the areas of the two rectangles?
iii. What operation is needed to find the area of the original figure?
iv. What is the difference in area between the two rectangles?
v. What do you notice about your answers to (a), (b), (c), and (d)?
vi. Why do you think this is true?

Example 2: Decomposing Polygons into Rectangles and Triangles
Parallelogram ABCD is part of a large solar power collector. The base measures 6 m and the height is 4 m.

a. Draw a diagonal from A to c. Find the area of both triangles ABC and ACD.
b. Draw in the other diagonal, from B to D. Find the area of both triangles ABD and BCD.

Example 3: Decomposing Trapezoids

The trapezoid below is a scale drawing of a garden plot.
Find the area of both triangles ABC and ACD. Then find the area of the trapezoid.

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Problem Set

1. If AB = 20 units, FE = 12 units, AF = 9 units, and DE = 12 units, find the length of the other two sides. Then, find the area of the irregular polygon.
2. If DC = 1.9 cm, FE = 5.6 cm, AF = 4.8 cm, and BC = 10.9 cm, find the length of the other two sides. Then, find the area of the irregular polygon.
3. Determine the area of the trapezoid below. The trapezoid is not drawn to scale.
4. Determine the area of the shaded isosceles trapezoid below. The image is not drawn to scale.
5. Here is a sketch of a wall that needs to be painted:
   a. The windows and door will not be painted. Calculate the area of the wall that will be painted.
   b. If a quart of Extra-Thick Gooey Sparkle paint covers 30 ft, how many quarts must be purchased for the painting job?

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