Mathematics · 3rd Grade · Shapes and Space: Geometry and Area · Weeks 19-27

Area of Rectilinear Figures

Finding the area of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts.

• Common Core State Standards
• CCSS.Math.Content.3.MD.C.7.d

About This Topic

CCSS.Math.Content.3.MD.C.7.d asks students to find the area of rectilinear figures by decomposing them into non-overlapping rectangles, finding each rectangle's area, and adding. This extends earlier area work to more complex shapes. The key conceptual move is that area is additive: the total area of a figure equals the sum of the areas of its non-overlapping parts.

Rectilinear figures appear in architecture, floor plans, and everyday spaces, contexts that make this topic applicable beyond the classroom. Students who can flexibly decompose shapes also develop spatial reasoning that supports geometry in later grades. There is typically more than one valid way to decompose a rectilinear figure, and recognizing that multiple strategies yield the same total area deepens understanding of additivity.

Active learning formats are particularly productive here because decomposition is a spatial reasoning challenge where peer comparison is genuinely informative. When students see that a classmate cut an L-shape differently and arrived at the same total area, they internalize why the additivity principle holds in a way that a single demonstrated solution cannot replicate.

Key Questions

1. Design a strategy to decompose a complex rectilinear figure into simpler rectangles.
2. Explain how the sum of the areas of the decomposed parts relates to the total area of the figure.
3. Critique different decomposition strategies for efficiency and accuracy.

Learning Objectives

• Calculate the area of rectilinear figures by decomposing them into non-overlapping rectangles.
• Explain how the sum of the areas of decomposed rectangles equals the total area of a rectilinear figure.
• Design a strategy to decompose a given rectilinear figure into at least two non-overlapping rectangles.
• Compare two different strategies for decomposing a rectilinear figure and justify which is more efficient.
• Critique a classmate's decomposition strategy for accuracy and completeness.

Before You Start

Area of Rectangles

Why: Students must be able to calculate the area of a single rectangle using the formula length × width before they can find the area of combined rectangles.

Introduction to Geometric Shapes

Why: Students need to be able to identify basic shapes like rectangles and understand concepts like sides and angles to work with rectilinear figures.

Key Vocabulary

rectilinear figure	A shape made up of only horizontal and vertical line segments. Think of shapes that look like they are made from straight lines meeting at right angles.
decompose	To break down a larger shape into smaller, simpler shapes. For rectilinear figures, we break them into smaller rectangles.
non-overlapping	Shapes that do not share any space. When you decompose a figure, the smaller rectangles must fit together perfectly without covering each other.
area	The amount of two-dimensional space a shape covers. We measure area in square units, like square inches or square centimeters.

Watch Out for These Misconceptions

Common Misconception

There is only one correct way to decompose a rectilinear figure.

What to Teach Instead

Students who learn one decomposition method may not realize that alternative cuts are equally valid. Deliberately presenting two student decompositions side by side and showing they yield identical total areas broadens strategic flexibility and provides a concrete illustration of area additivity.

Common Misconception

Area of irregular shapes must be found by counting every square unit.

What to Teach Instead

Students who learned area by counting unit squares sometimes revert to counting when faced with large rectilinear figures. Encouraging the habit of labeling dimensions and computing each rectangle's area as length times width makes the connection to multiplication explicit and produces a far more efficient strategy.

Common Misconception

Rectilinear figures are only L-shapes.

What to Teach Instead

Rectilinear figures include any polygon with all right angles, including T-shapes, U-shapes, step shapes, and more complex configurations. Exposing students to varied examples prevents over-generalizing from a single shape type and builds spatial reasoning that applies across all rectilinear configurations.

Active Learning Ideas

Inquiry Circle: Multiple Decompositions

Groups receive an L-shaped grid figure and must find at least two different ways to decompose it into rectangles. They calculate the total area using each decomposition, confirm both give the same answer, and present both strategies to the class with an explanation of why the totals match.

25 min·Small Groups

Gallery Walk: Decomposition Strategies

Post four rectilinear figures around the room. Students circulate and write their decomposition plan on sticky notes for each figure, showing where they would draw the dividing line without yet calculating the area. The class compares strategies posted for each figure.

25 min·Whole Class

Think-Pair-Share: Which Cut Is More Efficient?

Present a complex rectilinear figure. Students individually choose a decomposition and justify it to a partner. Partners evaluate whether one strategy requires fewer computation steps and discuss why someone might prefer a particular decomposition for a specific figure.

20 min·Pairs

Individual Practice: Architect's Floor Plan

Students receive a simple floor plan on grid paper showing two connected rooms in an L-shape. They label the dimensions, choose a decomposition, calculate the area of each part, and write the total area of the space with the equation they used.

15 min·Individual

Real-World Connections

• Architects and interior designers use rectilinear figures when drawing floor plans for houses and buildings. They must calculate the area of different rooms and spaces to determine how much flooring, paint, or furniture is needed.
• Construction workers use these concepts when measuring and cutting materials like wood or carpet for projects. Accurately calculating the area of irregularly shaped spaces ensures materials are not wasted.

Assessment Ideas

Exit Ticket

Provide students with a drawing of an L-shaped rectilinear figure. Ask them to: 1. Draw lines to decompose the figure into two non-overlapping rectangles. 2. Write the area of each smaller rectangle. 3. Write the total area of the figure.

Discussion Prompt

Present students with a complex rectilinear figure (e.g., a U-shape). Ask: 'How can we find the area of this shape? What are at least two different ways to break it down into smaller rectangles? Discuss with a partner: Are both ways correct? Which way seems easier to calculate?'

Quick Check

Draw a rectilinear figure on the board with multiple possible decomposition lines. Ask students to hold up fingers to indicate how many rectangles they would use to decompose it. Then, have them sketch one possible decomposition on a mini-whiteboard and share their calculation for the total area.

Frequently Asked Questions

How do you find the area of rectilinear figures in 3rd grade?

Draw a line that divides the shape into two non-overlapping rectangles. Calculate each rectangle's area using length times width. Add the two areas together. There is usually more than one valid place to draw the dividing line. Students should verify that their two pieces cover the entire original shape with no gaps and no overlaps before computing.

What is a rectilinear figure in 3rd grade math?

A rectilinear figure is a polygon with all right angles, meaning every corner is a square corner. L-shapes, T-shapes, and step shapes are common examples. They can always be divided into rectangles without leaving any gaps, which is what makes the decomposition strategy both possible and reliable.

Why do different decompositions of the same rectilinear figure give the same area?

Because area is additive: the total area of a shape is fixed regardless of how you divide it, as long as the pieces cover the whole shape exactly once with no overlap and no gaps. This is directly analogous to the way 7 can be broken into 3 + 4 or 5 + 2: the total stays the same even as the parts change.

How does active learning help students find area of rectilinear figures?

Comparing multiple decomposition strategies in small groups exposes students to spatial approaches they might not generate alone. When students see that a classmate cut an L-shape horizontally while they cut it vertically and both arrived at the same area, they experience the additivity principle firsthand rather than hearing it stated. Gallery walks build this comparative understanding efficiently across the whole class.