4.5.2 Area of Polygons

Compute areas of polygons (rectangles, parallelograms, trapezoids, regular polygons) using appropriate formulas.

定义

The area of a polygon is the measure of the two-dimensional space enclosed within its boundaries, expressed in square units. For different types of polygons, area is calculated using specific formulas based on their geometric properties:

Rectangle: A quadrilateral with four right angles. The area depends on the length and width of the sides.

Parallelogram: A quadrilateral with opposite sides parallel and equal. The area depends on the base and the perpendicular height from that base.

Trapezoid: A quadrilateral with exactly one pair of parallel sides (called bases). The area depends on both parallel sides and the perpendicular distance between them.

Regular Polygon: A polygon with all sides equal and all interior angles equal. The area can be calculated using the apothem (perpendicular distance from center to a side) and the perimeter, or by dividing it into congruent triangles from the center. \nThe key principle is that area is always measured as the product of linear dimensions, resulting in square units, and the height used must always be perpendicular to the base.

核心公式

• \(A_{\text{rectangle}} = l \times w\)
• \(A_{\text{parallelogram}} = b \times h\)
• \(A_{\text{trapezoid}} = \frac{1}{2}(b_1 + b_2) \times h\)
• \(A_{\text{regular polygon}} = \frac{1}{2} \times a \times p\)
• \(A_{\text{regular polygon}} = \frac{1}{2} \times n \times s^2 \times \cot\left(\frac{\pi}{n}\right)\)

易错点

• ⚠️ Using slant height instead of perpendicular height: Students often confuse the slant height of a trapezoid or parallelogram with the perpendicular height, leading to incorrect area calculations. The height must always be measured perpendicular to the base.
• ⚠️ Forgetting to divide by 2 in trapezoid formula: A common error is calculating \(A = (b_1 + b_2) \times h\) without the factor of \(\frac{1}{2}\), resulting in double the correct area.
• ⚠️ Misidentifying which dimension is the height in parallelograms: Students may use the length of a slanted side instead of the perpendicular distance between parallel sides, especially when the parallelogram is not a rectangle.
• ⚠️ Incorrect application of apothem formula for regular polygons: Confusing the apothem with the radius (distance from center to vertex) or using the wrong formula when converting between different representations of a regular polygon's area.