Math 6 H Periods 1, 6 & 7 (Wednesday)

Polygons 4-5

A polygon is a closed figure formed by joining segments—the sides of the polygons at that endpoints—the vertices of the polygon. Polygons are names according to the number of sides they have.

Triangle 3 sides
Quadrilateral 4 sides
Pentagon 5 sides
Hexagon 6 sides
Octagon 8 sides
Decagon 10 sides

A polygon is REGULAR if all its sides and all its angles are congruent.
A regular triangle is the equilateral triangle
A regular quadrilateral is the square.

To name a polygon we name its consecutive vertices in order.
A diagonal of a polygon is a segment joining two non consecutive vertices.

To find the perimeter of a polygon add all the lengths of its sides. The perimeter is the distance around the figure. Finding the perimeter of a parallelogram can be done by computing the sum of the lengths or by using the distributive property to obtain
For instance a parallelogram with sides 9 cm and 6 cm
has a perimeter of 9 + 6 + 9 + 6 = 30 cm
but you could calculate that by 2(9) + 2(6) = 18 + 12 = 30 cm or using the distributive property, even 2(9+6) = 2(15) = 30 cm

If you have a regular polygon you can simple multiple the side by the number of sides in the polygon
For example,
a quadrilateral with side 16.5 m has a perimeter of 4(16.5) = 66 m

The sum of the measures of the angles of any pentagon is 540 degrees. If it is a regular pentagon, what must be the measure of each angle of the regular pentagon? 540/5 = 108 degrees.

The sum of the measures of the angles of any pentagon is 540 degrees. How can you prove that? Draw your pentagon and then draw all the diagonals from ONE of the vertices. Count the number of triangles created. Three. How many degrees does a triangle have? 180. Multiply the number of triangles created by 180… 540 is your answer. IT works every time. So How could you create a general rule or formula for the sum of the measures of the angles of any polygon with n sides?

Practice drawing various polygons—now practice drawing all the diagonals for each of them. Can you determine a general rule for the number of diagonals that can be drawn for any polygon?


Circles 4-6
A circle is the set of all points in a plane at a given distance from a given point O called the center
A segment joining the center to a point on the circle is called a radius (plural: radii) of the circle. All radii of a given circle have the same length and this length is called the radius of the circle.

A segment joining two points on a circle is called a chord, and a chord passing through the center is a diameter of the circle. The ends of a diameter divide the circle into two semicircles. The length of a diameter is called the diameter of the circle.
The perimeter of a circle is called the circumference.

The quotient Circumference ÷ diameter can be showed to be the same for all circles. Regardless of their size. This quotient is denoted by a Greek letter
π no decimal gives π exactly 3.14159 is used. In our book 3.14 is concerned a fairly good approximation.

C ÷ d = π

Formulas
Let C = circumference d = diameter, and r = radius

Then
C = πd d = C ÷ π
C = 2π r r = C ÷ (2π)

A polygon is inscribed in a circle if all of its vertices are on the circle.
It can be shown that three points not on a line determine a circle. There is one and only one circle that passes through the three given points.