Composite Figures
Find the area and perimeter of composite shapes — figures made by combining or subtracting standard shapes — a critical skill for real-world design and construction.
Almost nothing in the real world is a single perfect rectangle or circle. A floor plan is a rectangle with notches cut out. A window might be a rectangle topped with a semicircle. A swimming pool might combine a rectangle and a half-circle. Composite figure math is the geometry of real-world design — if you can calculate area for basic shapes, combining them lets you handle virtually any floor plan, blueprint, or fabrication problem.
- Decompose composite figures into recognizable basic shapes
- Calculate the area of composite figures by adding or subtracting component areas
- Find the perimeter of composite figures by identifying the outer boundary only
Split the figure into parts. Calculate each part's area. Add them together.
Use when: the figure looks like shapes joined together.
Find area of the big surrounding shape. Subtract the missing piece(s).
Use when: the figure is a shape with holes or notches cut out.
Decomposing a Composite Figure
L-shaped room = one big rectangle MINUS one small rectangle in the corner
10 × 8 = 80 ft²
3 × 4 = 12 ft²
68 ft²
An L-shaped room has overall dimensions 12 ft × 9 ft, with a 4 ft × 5 ft corner cut out. Find the floor area.
- Large rectangle: \(12 \times 9 = 108\) ft²
- Cut-out corner: \(4 \times 5 = 20\) ft²
- L-shape area: \(108 - 20 = 88\) ft²
The floor area is 88 ft². Flooring cost at $3.50/ft² = $308.
A decorative window is a rectangle 3 ft wide × 4 ft tall topped with a semicircle of diameter 3 ft. Find the total glass area.
- Rectangle area: \(3 \times 4 = 12\) ft²
- Semicircle (radius = 1.5 ft): \(\frac{1}{2}\pi(1.5)^2 = \frac{1}{2}\pi(2.25) = 1.125\pi \approx 3.53\) ft²
- Total: \(12 + 3.53 \approx 15.53\) ft²
The window requires approximately 15.53 ft² of glass.
A backyard has a 30 ft × 20 ft patio. A circular pool with diameter 12 ft is cut into it. Find the area of the deck (patio minus pool).
- Patio area: \(30 \times 20 = 600\) ft²
- Pool area (radius = 6 ft): \(\pi(6)^2 = 36\pi \approx 113.1\) ft²
- Deck area: \(600 - 113.1 \approx 486.9\) ft²
The deck area is approximately 486.9 ft². Sealing at $2/ft² costs about $973.80.
- A cross shape is made of a 6 × 6 square in the center, with four 2 × 4 rectangles extending from each side. Find the total area.
- A 10 cm × 10 cm square has a circle of radius 3 cm removed from its center. Find the remaining area.
- Find the perimeter of the L-shaped room in Example 1 (12 × 9 overall, 4 × 5 corner cut).
▶ Show Answers
- Center: \(6 \times 6 = 36\); 4 rectangles: \(4 \times (2 \times 4) = 32\); Total: 68 units².
- \(100 - \pi(9) = 100 - 9\pi \approx 100 - 28.27 \approx\) 71.73 cm².
- Trace the outer edge: 12 + 9 + 4 + 5 + 8 + 4 = 42 ft. (Note: interior corner edges are NOT included.)
- Counting interior edges in the perimeter: Perimeter is the outer boundary only. When two shapes share an edge, that edge disappears from the perimeter — don't count it.
- Forgetting the ½ for semicircles: A semicircle is half a circle: area = \(\frac{1}{2}\pi r^2\). The diameter side is NOT part of the curved perimeter — add it separately if needed for the full perimeter.
- Applying the wrong strategy: "Addition" when shapes are joined together; "subtraction" when a piece is cut out. Choose based on the shape's structure.
- Decompose first: identify all the basic shapes — rectangles, triangles, circles, semicircles.
- Add areas when shapes are combined; subtract areas when a piece is removed.
- Perimeter = outer boundary only — shared interior edges vanish when shapes merge.
- Label dimensions clearly before calculating — composite problems are error-prone without a diagram.
Flooring contractors calculate composite floor areas every day: open-plan rooms with alcoves, kitchens with islands cut out, hallways that join larger spaces. They must add and subtract areas correctly to order the right amount of material — over-ordering wastes money; under-ordering causes delays. Interior designers calculate composite wall areas (subtracting windows and doors) to determine paint and wallpaper quantities. This is practical geometry at its most applied.
Calculator Connection
Use the Rectangle Calculator for rectangular components, the Circle Calculator for circular and semicircular portions, and the Trapezoid Calculator for trapezoidal sections. Combine results manually using addition or subtraction for the composite total.
Try it with the Calculator
Apply what you've learned with these tools.
Composite Figures — Quiz
5 questions per attempt · Intermediate · Pass at 70%
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