Question

How can you decompose the composite figure to determine its area?

Answer 1

I don’t see the figure you mean, so here’s a general, practical way to decompose any composite figure to find its area — with tips and short examples. If you can upload or describe the specific figure I’ll work it out step‑by‑step.

General strategy (step‑by‑step)
1. Identify simple shapes inside the composite: rectangles, triangles, circles (or sectors/semicircles), trapezoids, etc.
2. Draw auxiliary lines to split the figure into those simple shapes (horizontal/vertical cuts, diagonals, radii). Use symmetry where possible.
3. Compute the area of each simple piece using standard formulas. Common formulas:
- Rectangle: A = length × width
- Triangle: A = 1/2 × base × height
- Circle: A = πr^2 (semicircle = 1/2 πr^2, sector = 1/2 r^2 θ in radians)
- Trapezoid: A = 1/2 × (base1 + base2) × height
- Irregular polygon: split into triangles or use the shoelace formula
4. Add areas of pieces that are part of the figure; subtract areas of holes or cutouts. (Use inclusion–exclusion.)
5. Check for overlap or double counting; verify boundaries and units. If coordinates are available, you can place the figure on a grid and use coordinates or the shoelace formula for a polygon:
- Shoelace: A = 1/2 |Σ (x_i y_{i+1} − x_{i+1} y_i)| for ordered vertices.
6. State the final area with units and, if needed, exact form (e.g., 60 + (9/2)π) and decimal approximation.

Tips and common tricks
- Use symmetry: compute one half and double it.
- For curved boundaries, subtract or add circular segments/sectors.
- If shapes are inconvenient, extend to a larger simple shape and subtract the extra part.
- For a figure made of congruent parts, find area of one and multiply.
- For irregular boundaries, triangulate or use a grid count (useful for rough estimates).

Short examples
1) L‑shaped region: outer rectangle 8×6 with an inner 4×3 corner removed.
- Outer area = 8×6 = 48
- Removed area = 4×3 = 12
- Total area = 48 − 12 = 36 (square units)

2) Rectangle with a semicircle attached on one side: rectangle 10×6, semicircle radius 3 attached to a 6 side.
- Rectangle = 10×6 = 60
- Semicircle = 1/2 π r^2 = 1/2 π (3^2) = 4.5π
- Total area = 60 + 4.5π (≈ 74.14)

If you share the specific composite figure (image or dimensions/labels), I’ll decompose it and compute the exact area for you.