Students derive the formulas for the perimeter and area of regular polygons using the side and apothem lengths.
After completing this tutorial, you will be able to complete the following:
Recall that the perimeter of a region is the length of the path that surrounds a region (or the sum of the lengths of the sides of the region), and that the area of this region is the number of square units covered by the region. A regular polygon is a polygon whose sides and angles are all congruent. Since the perimeter of any polygon is the sum of its side lengths, the perimeter of an n-sided regular polygon with side length s is P = n ? s.
We can derive the formula for the area of a regular polygon from the formula for the area of a triangle or a rectangle. For the first derivation, consider the diagram of an n-sided regular polygon (in this case a pentagon, or five-sided polygon) in Figure 1.
In this diagram, the polygon is divided into n congruent triangles by drawing line segments from the center of the polygon (the point that is equidistant to the sides of the polygon) to its vertices. In addition, an apothem, or the shortest line segment from the center of the polygon to a side, is drawn in the bottom triangle. Suppose that the length of the apothem is a, and the length of a side of the polygon is s. Then, by the formula for the area of a triangle, the area of the bottom triangle isNow, since there are n congruent triangles, the area of the entire polygon isHowever, we already know that P = n ? s so we can rewrite this area formula using the perimeter:
In a regular hexagon (a six-sided polygon) with side length s, the line segments drawn from the center to the vertices divide the polygon into six congruent equilateral triangles. Since the height of an equilateral triangle with side length s is we have that the area of a regular hexagon is
Additionally, the formula for the area of a regular n-sided polygon can be derived from the formula for the area of a rectangle. For this derivation, consider the following polygon with perimeter P and apothem length a, and the rectangle formed from the triangles created by the line segments extending from the polygon's center to its vertices and the apothem in Figure 2. Notice that the length of the base of this rectangle is half the perimeter of the polygon and the height is the length of an apothem. So, the area of the corresponding rectangle is
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should be able to define a regular polygon and know its properties; know the definition of the center of a polygon; calculate the area of a parallelogram using base length and height; and calculate the area of a rectangle using length and width.|
|Type of Tutorial||Visual Proof|
|Key Vocabulary||apothem, area of regular polygon, perimeter of regular polygon|