## Finding the Area of Irregular Figures

Students will estimate the areas of highly irregular shapes and will use a process of decomposition to calculate the areas of irregular polygons.

The initial activity should be a free-draw for the students; that is, they should create a random shape on a blank piece of paper, such as the following:

To ensure that students do not draw a square, triangle, or some other common shape, you may wish to model this by drawing an irregular shape on the chalkboard or overhead projector.

Once all students have drawn a shape, ask them to estimate the area of their shape using any method they choose. Some students may overlay centimeter grid paper on top of the shape and then count squares. Others may draw squares, rectangles, and triangles within their shape and calculate the area of each polygon. Still others may compare their shape with other objects for which they know the exact dimensions and area, such as index cards or coins.

Allow students to compare their shape with a partner and discuss how they estimated the area. After sharing with a partner, have several students share their process with the entire class.

Next, distribute copies of the Polygons Activity Sheet, and have students cut out the shapes.

In their groups, ask students to create a design with their shapes. The design should have no overlaps or gaps; in other words, sides should touch to form an irregular figure. (Because students will be handling these figures and moving them around, you might want to copy them onto more durable paper.) Ask each group to discuss how they would determine the area of the figure that they created.

Rather than arranging paper shapes, students can instead use the Patch Tool to create a design. Note that the Patch Tool is mobile-friendly. A printed or saved copy can be used for measuring and calculating the area. (Alternatively, students can determine the area by considering the area of a triangle to be one unit, and all other shapes can be described in a relative manner; for instance, a blue parallelogram is equal to two triangles, so its area is two units.)

As a class, discuss the process of determining the area of each shape and then adding the areas. Students should calculate the area of each shape and record it directly on each shape, and then they should determine the total area of their figure by adding the area of all shapes that comprise their design.

Explain to students that *composition *is the act of putting
pieces together to form a whole, as they did with the shapes to create
a design. Conversely, *decomposition *is the act of breaking something
down into smaller parts, which is what they did to find the area of the
entire design. Tell them that they will use a process of decomposition
to determine the area of a larger space.

Identify a room in your school with an irregular shape (that is, the room should not be completely rectangular; if possible, there should be several angles that are not 90°.) Using that room as a basis, pose the following problem, or a similar one, to students:

Our principal wants to know how much carpet (or how many floor tiles) would be needed to cover the entire floor of this room. He needs to know the exact area, because he doesn’t want to order too much carpet and waste money, and he doesn’t want to order too little and not be able to cover the entire floor. Your job is to measure this room, determine the area of the floor, and write a letter to the principal telling him how much carpet to buy and how you arrived at your estimate.

If possible, take your students to the room around which the problem is based and let them measure the dimensions. If that is not possible, provide your students with a floor plan of the room. Alternatively, instead of using a room within your school, you could choose a famous room with irregular dimensions, such as the Octagon Room of the Royal Observatory in Greenwich, England. A different example is shown below.

In pairs, students should calculate the area of the floor. Using the ideas covered so far in this lesson, some pairs will naturally decompose the room into simpler polygons and calculate the individual areas. Other pairs will use other methods to estimate the area. After all pairs have arrived at an answer, conclude the lesson by letting students share their methods. Students who estimated will benefit from hearing about the decomposition method. Students who decomposed the room into simpler shapes will reinforce their estimation skills. Have students compare the results with one another. How close were the estimates to the actual area?

- Rulers, yardsticks, meter sticks, or measuring tapes Calculators
- Scissors
- Polygons Activity Sheet
- Centimeter grid paper (optional)
- Floor plan of a room in your school (optional)
- Computers or tablets with internet access (optional)

**Assessment Options**

- Have students determine the area of several U.S. states, such as California, Minnesota, Texas, Florida, or New York. These shapes are irregularly shaped, and their areas can easily be found in geography books or on the Internet. In addition to assessing student's ability to determine the area of an irregular figure, this activity would also provide practice with reading maps and using appropriate scales.
- Ask students to write a letter to the principal explaining how much carpet is needed and how they calculated the area.
- Allow students to use a ruler to create an irregular polygon with at least six sides. Then, have students exchange their drawings with a partner and determine the area of the figure they were given. Students should exchange back and verify their partner’s results.

**Extensions**

- When creating floor plans of the room in the school building, have students draw them to scale. Students could draw their plans on centimeter grid paper, for instance. They could include desks, shelves, and other items found in the rooms in their scale drawings.
- Have students create nets for prisms and pyramids. Ask students to find the surface area of these shapes using the method of decomposition.
- Have students determine the area of a shape that is drawn
within a rectangle, such as the pentagon contained in the rectangle
below.
In order to determine the area of the pentagon, students will have to subtract the area of the triangles from the area of the rectangle. [The area is 17.5 square units.]

**Questions for Students**

1. What techniques did you use to estimate the area of the irregular figure that you drew? What other methods could you have used?

[There are many possible methods. One is to draw the figure on grid paper, or overlay the figure with transparent grid paper, and count the squares. Another is to approximate the shape with known figures such as triangles and squares, determine the area of each, and add them.]

2. What real-life applications are there to what was done in this lesson? What kinds of jobs or tasks would require someone to complete an activity like the one that you did?

[Any occupation that requires knowing the area of a surface would benefit from knowing how to decompose a shape into smaller figures to compute the area. Flooring specialists, carpenters, painters, etc.]

**Teacher Reflection**

- When the students measured the dimensions of the room, how precise were their measurements? Did they experience any difficulties while measuring? What help did you need to offer students?
- Was your lesson appropriately adapted for the diverse learner?
- How did students demonstrate that they connected the area of the individual shapes to the area of the larger shape? What words did the students use to demonstrate that they understood the connection?
- How did your lesson address auditory, tactile and visual learning styles?

### Discovering the Area Formula for Triangles

### Finding the Area of Trapezoids

### Finding the Area of Parallelograms

### Learning Objectives

- Estimate the area of irregular shapes.
- Decompose irregular polygons into squares, rectangles, triangles, and other familiar shapes.
- Determine the area of an irregular polygon by summing the areas of its composite shapes.

### Common Core State Standards – Mathematics

Grade 7, Geometry

- CCSS.Math.Content.7.G.B.6

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.