Area of Composite (Rectilinear) Figures: Fifth Grade Mathematics
Composite or rectilinear figures are irregular shapes composed of multiple joined rectangles, whose total area is determined by additive decomposition or subtractive framing.
Method 1: The Additive Decomposition Strategy
To find the area of an irregular rectilinear shape using the additive strategy:
1. Divide the composite shape into two or more non-overlapping rectangles using a horizontal or vertical split line.
2. Determine the missing side lengths of each individual rectangle.
3. Calculate the area of each individual rectangle separately (Area = l * w).
4. Add all the individual areas together to find the total area.
+--------------------------------------------------------------------------+
| ADDITIVE DECOMPOSITION OF AN L-SHAPED FIGURE |
| |
| 8 cm |
| +-------------+ |
| | | |
| 10 cm| Rectangle A | 6 cm |
| | (8 * 6) | |
| +-------------+-------+ 4 cm |
| | Rectangle B | |
| | (14 * 4) | |
| +---------------------+ |
| 14 cm |
| |
| Rectangle A Area: 8 cm * 6 cm = 48 sq cm |
| Rectangle B Area: 14 cm * 4 cm = 56 sq cm |
| Total Area: 48 + 56 = 104 sq cm |
+--------------------------------------------------------------------------+
Method 2: The Subtractive (Framing) Strategy
Alternatively, enclose the entire figure inside a large bounding rectangle:
1. Find the area of the large outer bounding rectangle.
2. Find the area of the missing rectangular "cutout" space.
3. Subtract the cutout area from the large bounding area: Total Area = Outer Area - Cutout Area.
Subtractive Method for the L-Shape Above:
Large outer bounding rectangle: 14 cm wide by 10 cm high -> 14 * 10 = 140 sq cm.
Cutout rectangle in upper right:
Width = 14 - 8 = 6 cm.
Height = 10 - 4 = 6 cm.
Cutout Area = 6 * 6 = 36 sq cm.
Total Area = 140 - 36 = 104 sq cm. Both methods match!
Exercises and Solutions
Exercise 1: Decomposing a T-Shaped Figure. A wooden deck is shaped like a capital letter "T". The horizontal top beam of the T measures 12 feet long and 4 feet wide. The vertical stem of the T measures 6 feet long and 4 feet wide, attached to the center of the top beam. Find the total area of the deck.
Solution 1: Divide the T-shaped deck into two non-overlapping rectangles: Rectangle 1 (the top horizontal beam) and Rectangle 2 (the vertical stem). Find the area of Rectangle 1: Area 1 = 12 feet * 4 feet = 48 square feet. Find the area of Rectangle 2: Area 2 = 6 feet * 4 feet = 24 square feet. Add the two areas together: Total Area = 48 + 24 = 72 square feet.
Exercise 2: Deductions of Missing Sides in an L-Shape. An L-shaped room has an outer boundary with horizontal lengths of 15 feet and 9 feet, and vertical heights of 11 feet and 5 feet. Identify the missing dimensions, calculate the area using additive decomposition, and verify using the subtractive method.
Solution 2: Analyze dimensions: The total bottom horizontal width is 15 ft; the upper horizontal width is 9 ft, leaving the lower-step width as 15 - 9 = 6 ft. The total left vertical height is 11 ft; the lower-step height is 5 ft, leaving the upper vertical height as 11 - 5 = 6 ft. Using additive decomposition: split vertically into Left Rectangle (9 ft * 11 ft = 99 sq ft) and Right Rectangle (6 ft * 5 ft = 30 sq ft). Total Area = 99 + 30 = 129 square feet. Using subtractive method: bounding box is 15 ft * 11 ft = 165 sq ft. Missing cutout corner in upper right is 6 ft * 6 ft = 36 sq ft. Total Area = 165 - 36 = 129 square feet. Both methods confirm the area is 129 square feet.