The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area.
Perimeter: The distance around a shape.
Circumference: The distance around a circle.
Area: The amount of surface covered by a figure.
Center (of the polygon): The center of the circumscribed circle.
Radius (of the polygon): The radius of the circumscribed circle.
Apothem: A perpendicular segment from the center to a side of the polygon.
The perimeter is the sum of all the edges of a two-dimensional figure. The perimeter is measured in units of length (e.g. feet, inches, centimeter). If the unit is not specified, the perimeter is measured in “units.”
The circumference is also measured in units of length (e.g. feet, inches, centimeter). If the unit is not specified, the circumference is measured in “units.”
The area is measured in square units (e.g. square feet or \(ft^2\)). If the unit is not specified, the area is measured in “\(units^2\).”
A regular polygon is a polygon that is equiangular (all angles are equal) and equilateral (all sides have the same side lengths). All regular polygons can be inscribed in a circle, so these polygons also have a center and a radius.
The simplest way to find the perimeter of a regular polygon would be to just add the lengths of all the sides.
P = ns
There is another version of the perimeter formula: \(p = 2nr \sin (\frac{180^{\circ}}{n})\)
Theorem: Area = \(\frac{1}{2}nas = \frac{1}{2}pa\)
Can also be written as: Area = \(nr^2 \sin (\frac{180^{\circ}}{n}) \cos (\frac{180^{\circ}}{n})\)
Congruent Areas Postulate: If two figures are congruent, they have the same area.
If the polygon is not a regular polygon, the area can be found by dividing the polygon into smaller polygons where the areas can be calculated.
Area Addition Postulate: If a figure is composed of two or more parts that do not overlap each other, then the area of the figure is the sum of the areas of the parts.
If you remember the formulas for perimeter and area for rectangles and triangles, you can always divide other shapes into rectangles and triangles.
Perimeter of Similar Polygons Theorem: If two polygons are similar, then the ratio of the perimeters is equal to the ratio of the corresponding side lengths.
If \(\color{red}{ABCD} \color{black}{ \sim } \color{blue}{QRST}\), then \(\frac{\color{red}{AB + BC + CD + DA}}{\color{blue}{QR + RS + ST + TQ}} \color{black}{=} \frac{\color{red}{AB}}{\color{blue}{QR}} \color{black}{=} \frac{\color{red}{BC}}{\color{blue}{RS}} \color{black}{=} \frac{\color{red}{CD}}{\color{blue}{ST}} \color{black}{=} \frac{\color{red}{DA}}{\color{blue}{TQ}}\)
Area of Similar Polygons Theorem: If the scale factor of the sides of two similar polygons is \(\frac{m}{n}\), then the ratio of the
areas would be \((\frac{m}{n})^2\)
If \(\color{red}{ABCD} \color{black}{ \sim }\color{blue}{QRST}\) and \(\frac{\color{red}{AB}}{\color{blue}{QR}}\) is the scale factor, then the ratio of the areas is \((\frac{\color{red}{AB}}{\color{blue}{QR}})^{\color{black}{2}}\).