Area of Polygons and Circles
Area formulas can be found at “Reference Table for Areas”
Let’s pick up some hints for those more challenging problems involving area.
Regular polygons have a center and a radius (coinciding with their circumscribed circle), and the distance from the center perpendicular to any side is called its apothem.
The apothem of a regular polygon is a line segment from the center of the polygon perpendicular to any side of the polygon. Triangle DOC is an isosceles triangle, making the apothem the altitude of this triangle and the median of this triangle (going to the midpoint P.) The apothem is also the radius of the inscribed circle.
The apothem can be used to determine area:
Area of Sector of a Circle
Closed figures (i) and (ii) are different from (iii) and (iv). In figures (iii) and (iv) line segments cross themselves at one or more points. Thus, closed figures, which cross themselves at one or more points, are called complex closed figures, and the figures which do not cross themselves at any point are called simple closed figures. So, figures (i) and (ii) show simple closed figures and (iii) and (iv) show complex closed figures.
Each of these simple closed figures divides the plane into three parts. The first part is covered with cross (×), the second is the dotted part (•), and the third is the boundary of the figures. The part covered with crosses forms the interior of the figure. The part containing the dots, i.e., M, N, O, S, is called the exterior of the figures and the points A, B, C, D, E and P, Q, R lying on the boundary are called the points on the boundary of the figure.
