The document defines and explains properties of regular polygons. It states that regular polygons are polygons with all sides of equal length and all interior angles of equal measure, making them both equilateral and equiangular. It then lists nine properties of regular polygons, including that the center of the circumscribing circle, inscribed circle, and polygon coincide and that diagonals through the center are equal to the diameter. Formulas are provided for the area of each polygon segment, total area, perimeter, central angle, exterior angle, and interior angle of regular polygons. Examples are given to find the area of a regular pentagon and shaded region of an overlapping triangle configuration related to a regular hexagon.

1. The Regular Polygon
Collegeof EngineeringandComputerStudies,
St. Michael’s College
Iligan City

2. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Regular Polygon
Regular polygons are
polygons with all sides
equal and all included
angles equal. Meaning,
regular polygons are
both equilateral and
equiangular.

3. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Properties of Regular Polygon
1. The center of the circumscribing circle,
the center of inscribed circle, and the
center of polygon itself are coincidence.
2. All sides of regular polygon are equal in
length; it is denoted by x in the figure.
3. All included angles are equal; it is
denoted by β.
4. All external angles α, are equal.
5. Central angles of each segment are
equal; it is denoted by θ.

4. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Properties of Regular Polygon (cont’n.)
6. The apothem is the radius of the inscribed circle,
r.
7. The number of sides is equal to the number of
vertices, both are denoted by n.
8. Diagonals that pass through the center has
length equal to the diameter of the
circumscribing circle.
9. The triangular segment with area denoted as
A1 is an isosceles triangle. The length of the two
equal sides of this triangle is the radius of the
circumscribing circle and the altitude of this
triangle is the radius of the inscribed circle.

5. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Formulas of Regular Polygon
Area of one segment, A1
𝐴1 =
1
2
𝑥𝑟; 𝐴1 =
1
2
𝑅2 sin 𝜃
Total Area, A
𝐴 = 𝑛𝐴1
Perimeter, P
𝑃 = 𝑛𝑥
Central Angle, 𝜽
𝜃 =
360°
𝑛
Exterior angle, 𝜶
𝛼 = 𝜃
Interior angle, 𝜷
𝛽 = 180° − 𝛼
𝛽 = 180°
𝑛 − 2
𝑛

6. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Problems 1:
Find the area of the regular pentagon as shown in
the figure.

7. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Problems 2:
Three squares are drawn so that
each will contain a side of regular
hexagon as shown in the given
figure. If the hexagon has a
perimeter of 60 in., compute the
area of the region common to the
three squares. The required area
is the shaded region in the figure.

8. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Solution:

9. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Solution (cont’n.):