Decomposing Irregular and Rectilinear Figures - Third Grade Mathematics

Not all rooms, playgrounds, or swimming pools are simple plain rectangles. Many real-world spaces have L-shapes, T-shapes, or U-shapes with interesting corners. In mathematics, a shape whose sides all meet at right angles is called a rectilinear figure. While there is no single multiplication formula for an irregular rectilinear shape, you can easily find its area by decomposing (chopping) it into non-overlapping rectangles! In this chapter, you will master the art of decomposing rectilinear figures.

What Is a Rectilinear Figure?

A rectilinear figure is a polygon where every single angle is a right angle (90 degrees). All of its sides are either horizontal or vertical line segments.

Common Rectilinear Shapes:
L-Shape:             T-Shape:             U-Shape:
+-----+              +-----------+        +---+       +---+
|     |              |           |        |   |       |   |
|     +-------+      +---+   +---+        |   +-------+   |
|             |          |   |            |               |
+-------------+          +---+            +---------------+

Even though you cannot find the area of an L-shape in a single multiplication step, the shape is simply two regular rectangles joined together!

The Three-Step Decomposing Strategy

To find the area of any rectilinear figure:
1. Decompose: Draw a dashed straight line to slice the irregular figure into two (or more) non-overlapping rectangles.

2. Calculate: Find the area of each individual rectangle using Area = Length x Width.

3. Combine: Add the separate areas together to find the grand total area!


Formula: Total Area = Area of Rectangle A + Area of Rectangle B

Worked Example: Decomposing an L-Shape

Let us examine this L-shaped room:

           3 m
       +---------+
       |         |
   6 m |         | 4 m
       |         +---------------+
       |                         | 2 m
       +-------------------------+
                 8 m

Option 1: Vertical Slice

Draw a dashed vertical line going straight down from the inner corner:

           3 m         5 m
       +---------+---------------+
       |         |               |
   6 m | Rect. A |    Rect. B    | 2 m
       |         |               |
       +---------+---------------+
           3 m         5 m
  • Rectangle A (left): Width = 3 m, Height = 6 m. Area of A = 3 m x 6 m = 18 square meters.
  • Rectangle B (right): Height = 2 m. What is its length? The total bottom is 8 m, and Rectangle A uses 3 m. Length of B = 8 m - 3 m = 5 m! Area of B = 5 m x 2 m = 10 square meters.
  • Combine the areas: Total Area = 18 + 10 = 28 square meters!

Option 2: Horizontal Slice

What if you sliced horizontally instead?

           3 m
       +---------+
   4 m | Rect. C |
       +---------+---------------+
   2 m |       Rectangle D       | 2 m
       +-------------------------+
                 8 m
  • Rectangle C (top): Width = 3 m, Height = 4 m. Area of C = 3 m x 4 m = 12 square meters.
  • Rectangle D (bottom): Length = 8 m, Height = 2 m. Area of D = 8 m x 2 m = 16 square meters.
  • Combine the areas: Total Area = 12 + 16 = 28 square meters!

Notice that both slicing methods gave the exact same answer: 28 square meters! You can slice vertically or horizontally—whichever feels easiest for you.

Finding Hidden Side Dimensions

When decomposing rectilinear figures, some side lengths might not be directly labeled. You can find them by looking at opposite parallel sides:
- Horizontal Rule: The long horizontal bottom side equals the sum of the horizontal top segments!

- Vertical Rule: The long vertical side equals the sum of the vertical segments on the other side!


Chapter Practice Exercises

  1. Find the total area of each decomposed rectilinear figure: a. Figure split into Rectangle A (4 cm x 5 cm) and Rectangle B (3 cm x 2 cm) b. Figure split into Rectangle A (6 m x 8 m) and Rectangle B (4 m x 4 m) c. Figure split into three rectangles of areas 15 sq ft, 20 sq ft, and 12 sq ft
  2. An L-shaped garden is composed of two rectangular sections: Section 1 is 5 feet by 6 feet. Section 2 is 4 feet by 3 feet. What is the total area of the garden?
  3. A large rectangular piece of paper measures 10 inches by 8 inches. A small corner rectangle measuring 3 inches by 4 inches is cut out. What is the area of the remaining paper?
  4. Look at the L-shape from our worked example (Total Area = 28 sq meters, perimeter side lengths: 6 m, 3 m, 4 m, 5 m, 2 m, 8 m). What is the perimeter of the L-shape?
  5. Draw or describe an irregular rectilinear shape that has a total area of 20 square units. Show how you decompose it into two smaller rectangles.

Solutions and Step-by-Step Answers

  1. Total areas: a. Area A = 4 x 5 = 20 sq cm; Area B = 3 x 2 = 6 sq cm. Total = 20 + 6 = 26 sq cm. b. Area A = 6 x 8 = 48 sq m; Area B = 4 x 4 = 16 sq m. Total = 48 + 16 = 64 sq m. c. Three sections: 15 + 20 + 12 = 47 square feet.
  2. Garden area: Section 1: 5 x 6 = 30 sq ft. Section 2: 4 x 3 = 12 sq ft. Total area = 30 + 12 = 42 square feet.
  3. Subtraction method: Total original area: 10 in x 8 in = 80 square inches. Cut-out area: 3 in x 4 in = 12 square inches. Remaining area = 80 - 12 = 68 square inches.
  4. Perimeter of the L-shape: Add all outside edges: 6 + 3 + 4 + 5 + 2 + 8 = 28 meters! (Notice that in this special case, the perimeter is 28 m and the area is 28 sq m, but their units are completely different!).
  5. Sample shape: A 2 by 6 rectangle (area = 12 sq units) joined to a 2 by 4 rectangle (area = 8 sq units). Total area = 12 + 8 = 20 square units.