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 radius for a regular polygon is the same as the radius of the circumscribed circle.
- When the regular polygon is inscribed in a unit circle, the radius is 1.
- A central angle is the angle formed by two radii drawn to consecutive vertices of the polygon.
- The angle measure is \((\frac{360^{\circ}}{n})\)
- Length of an
**apothem**= \(r \cos (\frac{360^{\circ}}{n})\) - r is the length of the radius
- n is the number of sides

The simplest way to find the perimeter of a regular polygon would be to just add the lengths of all the sides.

P = ns

- n = number of sides
- s = side lengths

There is another version of the perimeter formula: \(p = 2nr \sin (\frac{180^{\circ}}{n})\)

- n = number of sides
- r = radius length

Theorem: Area = \(\frac{1}{2}nas = \frac{1}{2}pa\)

- P = perimeter
- a = apothem
- The area of the triangle is \(\frac{1}{2}as\), so the area of the regular polygon, which has n triangles, is n times the area of the triangle.

Can also be written as: Area = \(nr^2 \sin (\frac{180^{\circ}}{n}) \cos (\frac{180^{\circ}}{n})\)

- n = number of sides
- r = radius length

*Congruent Areas Postulate: *If two figures are congruent, they have the same area.

- The converse is not true! Two figures with the same area do not have to be congruent.

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.

*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 \((\frac{\color{red}{AB}}{\color{blue}{QR}})^{\color{black}{2}}\).