Regular Polygons

A regular polygon is a polygon that has all its sides and angles equal. The most common regular polygons are the equilateral triangle, square, and regular pentagon.

Equal sides: All sides of a regular polygon are of equal length.
Equal angles: All interior angles of a regular polygon are equal.
Sum of interior angles: The sum of the interior angles of a regular polygon with n sides is given by the formula: Sum = (n-2) * 180°.
Exterior angles: The measure of each exterior angle of a regular polygon is 360° / n, where n is the number of sides.

Here are some important formulas related to regular polygons:

The measure of each interior angle of a regular polygon with n sides is given by: Interior angle = (n-2) * 180° / n.
The apothem (distance from the center to the midpoint of a side) of a regular polygon can be calculated using the formula: Apothem = s / (2 * tan(180° / n)), where s is the length of a side.
The area of a regular polygon can be found using the formula: Area = (1/2) * Perimeter * Apothem.
The perimeter of a regular polygon is simply the sum of the lengths of all its sides.

Let's take an example of a regular pentagon with a side length of 6 units. We can use the above formulas to find its interior angle, apothem, area, and perimeter.

Number of sides, n = 5

Side length, s = 6 units

Interior angle = (5-2) * 180° / 5 = 108°

Using the formula for apothem:

Apothem = 6 / (2 * tan(180° / 5)) ≈ 3.08 units

Using the formula for area:

Area = (1/2) * Perimeter * Apothem = (1/2) * (5 * 6) * 3.08 ≈ 46.2 square units

When studying regular polygons, make sure to focus on the following key points:

Understanding the properties of regular polygons, including equal sides and angles.
Being able to calculate the interior and exterior angles of a regular polygon.
Mastering the formulas for finding the apothem, area, and perimeter of a regular polygon.
Practicing with examples to solidify your understanding of regular polygons.