# Determining the Area of Irregular Figures

Lesson 4 of 4

6th grade

75–90 minutes

**Description**

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

**Materials**

### Introduce

In partners or small groups, students should create an **irregular** **shape** on a blank piece of paper, such as the following.

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 sticky notes.

Allow students to work and 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.

### Explore

Distribute copies of the **Polygons** or Tangram Activity Sheet (download from Materials section above) and have students cut out the **shapes**.

In their groups, ask students to create an **irregular** **shape wi**ith their **shapes**. The **irregular** **shape** should have no overlaps or gaps; in other words, sides should touch to form an **irregular** **shape**. (Because students will be handling these **shapes** and moving them around, you might want to copy them onto more durable paper or use tangram sets.) Ask each group to determine the **area** of the irregular **shape** that they created.

Have students share out their **irregular** **shapes** and their method of determining the composite **area** of their figure.

### Synthesize

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?

[Guide them to come up with any relevant occupation, such as a flooring specialist, carpenter, painter, etc., that requires knowing the **area** of a surface would benefit from knowing how to decompose a **shape** into smaller figures to compute the **area**.]

Discuss possible situations when is it ok to estimate and when it is necessary to be exact.

### Assessment (optional)

Choose a famous room with **irregular** dimensions, such as the Octagon Room of the Royal Observatory in Greenwich, England.

Have students calculate the **area** of the floor. Some will naturally decompose the room into familiar **polygons** and calculate the individual **areas**, whereas others may use a different strategy to estimate the **area**. Have students justify their answer by explaining their strategy and why they chose to either estimate or find the exact **area**.

### Extension

**Activity 1**

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.

**Activity 2**

Have students determine the **area** of a **shape** that is drawn within a rectangle, such as the pentagon contained in the rectangle below.

**Activity 3**

Identify a room in your school with an **irregular** **shape** (that is, the room should not be completely rectangular; if possible, there should be at least one angle that is not 90 degrees.) If needed, provide a drawing of the layout of the room, with all lengths labeled. 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. Your job is to measure this room, determine the **area** of the floor, and write a letter to the principal telling him/ her how much carpet to buy and how you arrived at your answer. [Discuss with your students why accuracy is important.] The principal needs to know the exact **area**, because he/she doesn't want to order too much carpet and waste money, nor she does he/she want to order too little and not be able to cover the entire floor.

### Teacher Reflection

- How did students demonstrate that they connected the
**area** of the individual **shapes** to the **area** of the larger **shape**? How was the vocabulary the students were using to communicate with each other important to them being able to demonstrate that they understood the connection to the **area of** **irregular** **shapes**?
- How did students demonstrate their understanding of finding the
**area** of **triangles**, **polygons**, and **quadrilaterals** to help them determine the **area** of an **irregular** figure?
- What differentiation and scaffolding did you provide to ensure student success in connecting previous learned strategies to finding the
**area** of **irregular** figures?
- When the students measured the dimensions of the room, how precise were their measurements? What difficulties did students experience while measuring? What help did you need to offer students?

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