Perimeter & Area • Topic 3 of 3

3: Composite Figures and Mixed Problems

What are Composite Figures?

Composite figures (or compound shapes) are shapes made up of two or more basic geometric shapes (rectangles, squares, triangles, trapeziums, etc.) combined together.

Strategy for Finding Area of Composite Figures:

StepAction
1Divide the composite figure into simpler shapes
2Calculate the area of each simple shape separately
3Add areas for shapes that are combined
4Subtract areas for holes/shapes that are cut out
5Write final answer with correct units

Strategy for Finding Perimeter of Composite Figures:

  • Trace the outer boundary of the figure carefully
  • Add the lengths of all outer sides
  • Do NOT include internal lines where shapes join

Common Composite Figure Types:

TypeApproach
L-shape (two rectangles)Split into two rectangles, add areas
Rectangle with triangle on topArea of rectangle + area of triangle
Circle cut out of rectangleArea of rectangle - area of circle
Trapezium + triangleAdd both areas
Irregular polygonDivide into triangles and rectangles

Real-life Applications:

  • Floor plans of L-shaped rooms
  • Gardens with rectangular lawn and triangular flower bed
  • Construction of houses and buildings
  • Land area calculation for irregular plots
  • Design of logos and emblems
Composite Figures — Area of Complex ShapesRectangle 1Rect 2L-shaped = Rect1 + Rect2= (200×50)+(100×100)HollowAnnulus = π(R²−r²)= π(80²−40²) / scaleStrategy: Split complex shape → calculate parts → ADD or SUBTRACT
1
Worked Example
Example 1: Find the area of an L-shaped figure formed by two rectangles: one rectangle is 12 cm by 5 cm, the other is 8 cm by 4 cm attached to its side. (Assume they share a 5 cm side)
Solution- Step 1: Draw and label: Rectangle A: 12 × 5, Rectangle B: 8 × 4 attached along a 5 cm side - Step 2: Area of Rectangle A = \(12 \times 5 = 60\) cm² - Step 3: Area of Rectangle B = \(8 \times 4 = 32\) cm² - Step 4: Total area = \(60 + 32 = 92\) cm² - Step 5: For perimeter, trace outer boundary carefully - Step 6: Perimeter = \(12 + 5 + 8 + 4 + (12-8=4) + (5+4=9?)\) - Need careful addition: - Step 7: Let's trace: Start at top left: 12 (top) + 5 (right of rect A) + 8 (top of rect B) + 4 (right of rect B) + (12-8=4) (bottom horizontal) + (5+4=9) (left side) = 12+5+8+4+4+9 = 42 cm
2
Worked Example
Example 2: A rectangular garden 20 m long and 15 m wide has a square flower bed of side 5 m in one corner. Find the area of the remaining grass.
Solution- Step 1: Area of garden (rectangle) = \(20 \times 15 = 300\) m² - Step 2: Area of flower bed (square) = \(5 \times 5 = 25\) m² - Step 3: Remaining area = Garden area - Flower bed area - Step 4: \(300 - 25 = 275\) m²
3
Worked Example
Example 3 (Challenging): A window is in the shape of a rectangle with a semicircle on top. The rectangle is 1.4 m wide and 2 m tall. The semicircle has the same width as the rectangle (diameter = 1.4 m). Find the total area and perimeter of the window.
Solution- Step 1: Rectangle area = width × height = \(1.4 \times 2 = 2.8\) m² - Step 2: Semicircle radius = \(1.4 \div 2 = 0.7\) m - Step 3: Full circle area = \(\pi r^2 = 3.14 \times (0.7)^2 = 3.14 \times 0.49 = 1.5386\) m² - Step 4: Semicircle area = half of circle = \(1.5386 \div 2 = 0.7693\) m² - Step 5: Total area = \(2.8 + 0.7693 = 3.5693\) ≈ 3.57 m² - Step 6: Perimeter = bottom edge (1.4 m) + two sides (2 m + 2 m = 4 m) + semicircular arc - Step 7: Arc length = half of circumference = \(\pi \times d \div 2 = 3.14 \times 1.4 \div 2 = 4.396 \div 2 = 2.198\) m - Step 8: Total perimeter = \(1.4 + 4 + 2.198 = 7.598\) m

Key Points

  • Composite figures are made of two or more basic shapes
  • Break composite shapes into rectangles, squares, triangles, trapeziums, etc.
  • Area: Add areas of all parts (or subtract for holes)
  • Perimeter: Trace the outer boundary, add all external side lengths
  • Don't include internal lines in perimeter calculation
  • Always include proper units (cm², m² for area; cm, m for perimeter)
  • Draw a diagram and label all measurements before calculating
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Tap an option to check your answer0 / 4
Q1.The area of a composite figure is the ___ of the areas of its parts.
Explanation: Add the parts.
Q2.To find a composite area, split it into:
Explanation: Known shapes.
Q3.The area of a semicircle is:
Explanation: Half a circle.
Q4.To find a remaining area, you ___ the removed region.
Explanation: Subtract it.