Area of a regular polygon

Definition

The area of a regular polygon can be calculated using the formula: $$A = \frac{1}{2} \times P \times a$$, where $A$ is the area, $P$ is the perimeter, and $a$ is the apothem. This formula emphasizes the relationship between the polygon's side lengths and height, allowing for efficient computation of the area for any regular polygon, which is essential when dealing with complex geometric shapes.

5 Must Know Facts For Your Next Test

1. The apothem can be found using trigonometric functions when the length of a side and the number of sides are known.
2. Regular polygons include shapes like equilateral triangles, squares, and regular pentagons, all of which can be analyzed using this area formula.
3. This formula is derived from dividing the polygon into isosceles triangles, each having a vertex angle equal to the angle at the center of the polygon.
4. The area formula can be used to find areas of composite figures made up of multiple regular polygons by calculating individual areas and summing them up.
5. Knowing either the perimeter or the apothem allows you to find the area easily, making this formula versatile in geometric problem-solving.

Review Questions

• How does understanding the relationship between perimeter and apothem enhance your ability to calculate the area of various regular polygons?
  • Understanding that both perimeter and apothem are essential components in calculating the area allows you to see how changing one variable affects the overall area. For instance, if you know a polygon's perimeter and need to find its area, you can easily compute it by determining its apothem. This interrelationship simplifies calculations, especially when dealing with different types of regular polygons.
• Describe how you would use the area formula for a regular polygon to solve a problem involving a composite figure made up of multiple regular polygons.
  • To solve for the area of a composite figure made up of multiple regular polygons, you would first identify each individual polygon within the figure. Then, using the area formula $$A = \frac{1}{2} \times P \times a$$ for each polygon, calculate their respective areas by determining their perimeters and apothems. Finally, sum all individual areas to find the total area of the composite figure.
• Evaluate how changes in side length and number of sides affect both the perimeter and apothem of a regular polygon, thereby impacting its area.
  • Changes in side length directly affect both perimeter and apothem because as you increase side length while keeping the number of sides constant, you increase the perimeter linearly. However, increasing side length also affects how tall the apothem is relative to that side; generally, larger side lengths yield larger apothems too. Therefore, as both perimeter and apothem increase with side length, you see an exponential effect on area since it is calculated through multiplication in the formula $$A = \frac{1}{2} \times P \times a$$.

Related terms

Regular Polygon: A polygon with all sides and angles equal.

Perimeter:

The total distance around a polygon, calculated by adding the lengths of all its sides.

Apothem:

The perpendicular distance from the center of a regular polygon to the midpoint of one of its sides.