Regular Polygons

Let us learn about Regular Polygons,
There is really no limit to the number of sides a polygon may have. The only practical limit is that unless you draw them on a very large sheet of paper, after about 20 sides or so, the polygon begins to look very much like a circle.

Parts of a regular polygon
In a regular polygon, there is one point in its interior that is equidistant from its vertices. This point is called the center of the regular polygon. In Figure 1, O is the center of the regular polygon.

Figure 1 Center, radius, and apothem of a regular polygon.

The radius of a regular polygon is a segment that goes from the center to any vertex of the regular polygon.

The apothem of a regular polygon is any segment that goes from the center and is perpendicular to one of the polygon's sides. In Figure 1 , OC is a radius and OX is an apothem.

Finding the Perimeter

Because a regular polygon is equilateral, to find its perimeter you need to know only the length of one of its sides and multiply that by the number of sides. Using n-gon to represent a polygon with n sides, and s as the length of each side, produces the following formula.

Finding the Area

If p represents the perimeter of the regular polygon and a represents the length of its apothem, the following formula can eventually be shown to represent its area.

Example 1: Find the perimeter and area of the regular pentagon in figure 2 , with apothem approximately 5.5 in.

Figure 2

Finding the perimeter and area of a regular pentagon.

Hope the above explanation helped you.

Ratio and Proportion

Let us learn about ratio and proportion,
Ratio is a concept that you have probably encountered in other math classes. It is a comparison of sizes.

Ratio :

The ratio of two numbers a and b is the fraction , usually expressed in reduced form. An alternative form involves a colon. The colon form is most frequently used when comparing three or more numbers to each other. See Table 1 .

TABLE 1 Ratio Formats Ratio Written Form 3 to 4 3/4 or 3 : 4 a to b, b ≠ 0 a/b or a : b 1 to 3 to 5 1 : 3 : 5

Example 1: A classroom has 25 boys and 15 girls. What is the ratio of boys to girls?




The ratio of boys to girls is 5 to 3, or 5/3, or 5 : 3.
Example 2: The ratio of two supplementary angles is 2 to 3. Find the measure of each angle.

• The angles have measures of 72° and 108°.
  
• 2 x to 3 x reduces to 2 to 3.
  
• 2 x + 3 x = 180° (The sum of supplementary angles is 180°.)
  
• 5 x = 180°
  
• x = 36°
  
• Then, 2 x = 2(36°) and 3 x = 3(36°).
  
• So, 2 x = 72° and 3 x = 108°

The angles have measures of 72° and 108°.

Hope the above explanation helped.

Perimeters and Areas of similar triangles

Let us learn about perimeters and areas of similar triangles,
When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangles. In Figure 1 , Δ ABC∼ Δ DEF.

Figure 1 Similar triangles whose scale factor is 2 : 1.

The ratios of corresponding sides are 6/3, 8/4, 10/5. These all reduce to 2/1. It is then said that the scale factor of these two similar triangles is 2 : 1.
The perimeter of Δ ABC is 24 inches, and the perimeter of Δ DEF is 12 inches. When you compare the ratios of the perimeters of these similar triangles, you also get 2 : 1. This leads to the following theorem.
Theorem : If two similar triangles have a scale factor of a : b, then the ratio of their perimeters is a : b.
Hope the above explanation helped you.

Classifying Triangles by Sides or Angles

Let us study classifying triangles by sides or angles,
Triangles can be classified either according to their sides or according to their angles. All of each may be of different or the same sizes; any two sides or angles may be of the same size; there may be one distinctive angle.


The types of triangles classified by their sides are the following:



1. Equilateral triangle: A triangle with all three sides equal in measure. the slash marks indicate equal measure.

2. Isosceles triangle: A triangle in which at least two sides have equal measure

3. Scalene triangle: A triangle with all three sides of different measures


The types of triangles classified by their angles include the following:





1. Right triangle: A triangle that has a right angle in its interior

2. Obtuse triangle: A triangle having an obtuse angle (greater than 90° but less than 180°) in its interior.

3. Acute triangle: A triangle having all acute angles (less than 90°) in its interior

4. Equiangular triangle: A triangle having all angles of equal measure.

Hope the above explanation helped you, now let me explain polygons.