1. Understanding Composite Figures
A composite figure (or compound shape) is a shape made by combining two or more basic geometric figures such as rectangles, triangles, circles, semicircles, and trapeziums.
To find the area of such shapes, we break the figure into simpler known shapes, calculate their areas separately, and then add or subtract them depending on the situation.
This method works because area is a measure of 2D space, and areas of non-overlapping regions can be combined directly.
1.1. Idea Behind the Method
The key idea is:
- Divide the figure into known shapes (rectangle, triangle, circle, etc.)
- Calculate their areas with formulas you know
- Add areas if the shapes are joined
- Subtract areas if part of the figure is removed (like a hole)
This approach helps solve even very complex-looking shapes easily.
2. Steps to Solve Composite Area Problems
To calculate the area quickly and correctly, follow these steps:
2.1. Step-by-Step Method
- Observe the figure: Identify all basic shapes present.
- Break the figure: Sketch faint lines to divide it into rectangles, triangles, circles, etc.
- Label dimensions: Add lengths, breadths, radii, and heights to each shape.
- Apply formulas: Use known area formulas like:
- Rectangle: \(A = l \times b\)
- Triangle: \(A = \dfrac{1}{2}bh\)
- Circle: \(A = \pi r^2\)
- Trapezium: \(A = \dfrac{1}{2}(a + b)h\)
- Add or subtract: Combine the areas based on how the shapes are arranged.
3. Common Composite Shape Patterns
Most composite figures fall into a few common patterns that you will see often in textbooks and exams.
3.1. Rectangle + Semicircle
A semicircle attached to one side of a rectangle forms a common composite figure (e.g., archways, playground shapes).
The total area is:
\( A = A_{\text{rectangle}} + \dfrac{1}{2}\pi r^2 \)
3.2. Rectangle + Triangle
This shape appears in roof designs, signboards, and many diagram-based questions.
Combine the area formulas:
\( A = l \times b + \dfrac{1}{2}bh \)
3.3. Circle with a Removed Inner Circle (Ring or Annulus)
If a smaller circle is cut from a larger circle, the area becomes:
\( A = \pi (R^2 - r^2) \)
This is used in pipe cross-sections, rings, and circular borders.
4. Examples
Let’s solve some common types of composite figures using simple calculations.
4.1. Example 1: Rectangle + Semicircle
A rectangular playground is 20 m long and 10 m wide. A semicircle with radius 5 m is attached to one 10 m side. Find the total area.
Step 1: Area of Rectangle
\( A_1 = 20 \times 10 = 200\,\text{m}^2 \)
Step 2: Area of Semicircle
\( A_2 = \dfrac{1}{2}\pi r^2 = \dfrac{1}{2}\pi (5^2) = \dfrac{25\pi}{2} \)
Using \(\pi = 3.14\):
\( A_2 = \dfrac{25 \times 3.14}{2} = 39.25\,\text{m}^2 \)
Total Area:
\( A = 200 + 39.25 = 239.25\,\text{m}^2 \)
4.2. Example 2: Rectangle + Triangle
A figure consists of a rectangle of length 12 cm and breadth 8 cm, with a triangle on top having base 12 cm and height 6 cm.
Step 1: Area of Rectangle
\( A_1 = 12 \times 8 = 96\,\text{cm}^2 \)
Step 2: Area of Triangle
\( A_2 = \dfrac{1}{2} \times 12 \times 6 = 36\,\text{cm}^2 \)
Total Area:
\( A = 96 + 36 = 132\,\text{cm}^2 \)