4.5.4 Area of Composite Figures

Find areas of composite shapes by decomposing them into simpler geometric figures and summing their areas.

定义

A composite figure is a shape composed of two or more simple geometric figures (such as rectangles, triangles, circles, trapezoids, or parallelograms) combined together. To find the area of a composite figure, decompose it into non-overlapping simpler shapes, calculate the area of each individual shape using appropriate formulas, and then sum all the areas together. Alternatively, if the composite figure is formed by removing a smaller shape from a larger shape, subtract the area of the removed portion from the total area. This method is expressed as: \(A_{\text{composite}} = A_1 + A_2 + A_3 + \cdots + A_n\) (for addition) or \(A_{\text{composite}} = A_{\text{total}} - A_{\text{removed}}\) (for subtraction).

核心公式

• \(A_{\text{composite}} = A_1 + A_2 + A_3 + \cdots + A_n\)
• \(A_\text{rectangle} = l \times w\)
• \(A_\text{triangle} = \frac{1}{2} \times b \times h\)
• \(A_\text{circle} = \pi r^2\)
• \(A_\text{trapezoid} = \frac{1}{2}(b_1 + b_2) \times h\)

易错点

• ⚠️ Forgetting to decompose the figure correctly or missing one of the component shapes, leading to an incomplete calculation of the total area
• ⚠️ Incorrectly identifying which dimensions correspond to which shapes, especially when shapes share common sides or boundaries
• ⚠️ Using the wrong formula for a particular shape (e.g., using the full circle area formula instead of a sector formula when only part of a circle is included)
• ⚠️ Failing to recognize when subtraction is needed instead of addition, such as when a shape has a hole or cutout that must be removed from the total area