Area of composite figures (see also 5.4) Knowledge-Based Study

Study notes

1. What is a Composite Figure?

   A composite figure is a geometric shape made up of two or more basic two-dimensional shapes, such as rectangles, squares, triangles, parallelograms, trapezoids, or semicircles.

   To calculate the total area of a composite figure, we break the complex shape down into simpler components whose areas we can calculate directly using standard geometric formulas.

   Key Formulas for Component Shapes

   • Rectangle: $\text{Area} = \text{length} \times \text{width} = l \times w$
   • Square: $\text{Area} = \text{side}^2 = s^2$
   • Triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} b h$
   • Trapezoid: $\text{Area} = \frac{1}{2} (b_1 + b_2) h$
   • Parallelogram: $\text{Area} = b \times h$
   • Semicircle: $\text{Area} = \frac{1}{2} \pi r^2$

   Method 1: Decomposing into Simpler Parts

   The most common way to determine the area of a composite figure is by splitting it into non-overlapping regions and adding their areas together.

   Step-by-Step Strategy:

   1. Divide: Draw auxiliary lines to separate the figure into familiar, basic shapes.
   2. Find Missing Dimensions: Use the given side lengths to determine any unknown base, height, or radius values.
   3. Calculate Individual Areas: Apply the appropriate formula to each individual part.
   4. Sum the Areas: Add the calculated areas together to find the total area.

   Worked Example: L-Shaped Polygon

   Consider an L-shaped region with a vertical height of 10 cm, an overall base width of 8 cm, an upper horizontal edge of 3 cm, and a right vertical edge of 4 cm.

   • We can split this figure vertically into two rectangles:
     • Rectangle 1 (left): width = 3 cm, height = 10 cm
       $\text{Area}_1 = 3 \times 10 = 30\text{ cm}^2$
     • Rectangle 2 (right): width = $8 - 3 = 5\text{ cm}$, height = 4 cm
       $\text{Area}_2 = 5 \times 4 = 20\text{ cm}^2$
   • Total Area: $\text{Area} = \text{Area}_1 + \text{Area}_2 = 30 + 20 = 50\text{ cm}^2$.

   Method 2: Subtracting Unwanted Regions

   Sometimes, it is faster to place the composite figure inside a larger, regular bounding shape and subtract the areas of the regions that are not part of the figure.

   Step-by-Step Strategy:

   1. Enclose: Construct a larger bounding rectangle or square that completely contains the figure.
   2. Calculate Bounding Area: Find the area of the entire outer shape.
   3. Identify the Cutouts: Determine the dimensions and areas of the empty corners or holes.
   4. Subtract: Subtract the empty areas from the total bounding area.

   Worked Example: Re-evaluating the L-Shape

   Using the same L-shaped figure from the previous lesson:

   • Bounding Rectangle: base = 8 cm, height = 10 cm
     $\text{Total Area} = 8 \times 10 = 80\text{ cm}^2$
   • Cutout Corner: base = $8 - 3 = 5\text{ cm}$, height = $10 - 4 = 6\text{ cm}$
     $\text{Cutout Area} = 5 \times 6 = 30\text{ cm}^2$
   • Actual Area: $\text{Area} = 80 - 30 = 50\text{ cm}^2$.

   Both methods yield the exact same result. You can select whichever method requires fewer calculations.

   Figures with Semicircles and Triangles

   Composite figures frequently combine straight-edged shapes with curved components or inclined roofs.

   Worked Example: A Rectangular Wall with a Triangular Roof

   A figure is composed of a rectangle with a width of 12 m and a height of 7 m, topped by an attached triangle whose base is the top edge of the rectangle (12 m) and whose height is 5 m.

   1. Area of the rectangular base:
      $$\text{Area}_{\text{rect}} = 12 \times 7 = 84\text{ m}^2$$
   2. Area of the triangular top:
      $$\text{Area}_{\text{tri}} = \frac{1}{2} \times 12 \times 5 = 30\text{ m}^2$$
   3. Total composite area:
      $$\text{Area}_{\text{total}} = 84 + 30 = 114\text{ m}^2$$

   Common Pitfalls to Avoid

   • Double-counting shared boundaries: Remember that interior dividing lines are not boundaries of the total perimeter and must not introduce extra areas.
   • Diameter vs. Radius: When a semicircle is attached to an edge, that edge represents the diameter. Always divide by 2 to get the radius before using $\frac{1}{2}\pi r^2$.
   • Consistent Units: Ensure all linear dimensions are in the same unit before performing multiplications.