Name “N-GON” Circle

# of sides 3 4 5 6 8 n NA

Measure Central Angle NA

Measure internal angle NA

Side length in terms of radius

NA

Perimeter (in terms of radius)

Apothem (a) (in terms of radius) .80802r .9239r

Area (in terms of radius)

Name Equilateral Triangle

Square Pentagon Hexagon Octagon “N-GON” Circle

# of sides 3 4 5 6 8 n NA

Measure Central Angle NA

Measure internal angle NA

Side length in terms of radius

NA

Perimeter (in terms of radius) 2r*Π

Apothem (a) (in terms of radius) .80802r .9239r r

Area (in terms of radius) Πr^2

Triangles Central Angle

Interior Angle

Triangles

r

r

apothem

NameEquilateral Triangle

Square Pentagon Hexagon Octagon “N-GON” Circle

# of sides 3 4 5 6 8 n NA

Measure Central Angle 120 90 NA

Measure internal angle 60 90 NA

Side length in terms of radius r√3

= 1.73rr√2

=1.41rNA

Perimeter (in terms of radius) 4r√3

=5.20r4 r√2

=5.66r2r*Π

Apothem (a) (in terms of radius)

r/2=.5r

r√2/2.71r

.80802r .9239r R

Area (in terms of radius) r2 (3/4)√3

=1.30 r2 2 r2 Πr2

=3.14r2

Triangles

Find: Central Angle

Whole circle is 3600

There are 3 angles that make up 3600 thus Measure of central angle is = 360/3

Central Angle of triangle =1200

Triangles To find: Interior AngleMeasure of central=120Sum of all angles in the blue triangle is 180

each side angle will be = (180-120)/2 = 300

Interior Angle

Interior Angle= 2* 300

= 600

Triangles

r

r/2

r√3/2r

apothem 30

60

Side Length = 2 * r√3/2 = r√3

Perimeter = 3*side length = 3r√3

Triangles

Total Area = 3*Area of ORANGE TRIANGLE

Area ORANGE TRIANGLE= ½ b*a = ½ r√3 *r/2 = (r2/4)√3

= r2 (3/4)√3

r√3

r/2

SquareFind: Central Angle

Whole circle is 3600

There are 4 angles that make up 3600 thus Measure of central angle is = 360/4 = 900

Square

Find: Internal Angle

= 2* [180-90]/2 = 180-90 =

900

Square

r

r

r/√2

r/√2= r√2/2

45

45

Apothem = r√2/2

Side Length = 2 * r√2/2 = r√2

Perimeter = 4*side length = 4r√2

Square

45

45

r√2

r√2/2

Total Area = 4*Area of BLUE TRIANGLE

Area BLUE TRIANGLE= ½ b*a = ½ r√2 *r√2/2 = (r2/2)

= 2r2

r

Apothem= .80802r

r

Apothem= . 9239r

r

Apothem

NameEquilateral Triangle

Square Pentagon Hexagon Octagon “N-GON” Circle

# of sides 3 4 5 6 8 n NA

Measure Central Angle

120 90 72 60 45 NA

Measure internal angle

60 90 108 120 135 NA

Side length in terms of radius

r√3 = 1.73r

r√2=1.41r

1.1756r r .76537r NA

Perimeter (in terms of radius)

4r√3=5.20r

4 r√2=5.66r

5.8778r 6r 6.123r 2r*Π

Apothem (a) (of radius)

r/2=.5r

r√2/2.71r

.80802rr√3/2

= .866r.9239r R

Area (in terms of radius)

r2 (3/4)√3=1.30 r2 2 r2 2.378 r2

3√3/2r2

=2.598r22.828 r2 Πr2

=3.14r2

http://www.mathopenref.com/polygonapothem.html

Name Equilateral Triangle

Square Pentagon Hexagon Octagon “N-GON” Circle

# of sides 3 4 5 6 8 N NA

Measure Central Angle

120 90 72 60 45 360/n NA

Measure internal angle

60 90 108 120 135180(n-2)

nNA

Side length in terms of radius

r√3 = 1.73r

r√2=1.41r

1.1756r r .76537r 2√(r2 - a2) NA

Perimeter (in terms of radius)

4r√3=5.20r

4 r√2=5.66r

5.8778r 6r 6.123rn*side=

2n√(r2 - a2) 2r*Π

Apothem (a) (of radius)

r/2=.5r

r√2/2.71r

.80802rr√3/2

.865.9239r

Find out after trig!

R

Area (in terms of radius)

r2 (3/4)√3=1.30 r2 2 r2 2.378 r2

(3√3/2)r2

=2.598r22.828 r2 ½ a*p

Πr2

=3.14r2

Ex 1.) Find the area of the pentagon with apothem=3 & radius 5.

Ex 2.) Find the area of the hexagon with side length = 4.

4

Ex 3.) Find the shaded & un-shaded region of the equilateral triangle with side length 2 inscribed in the circle.

Ex 4.) Given 3 shapes: A square with diagonal length = 6, A regular hexagon with diagonal length= 6, and a circle with diameter equal to 6. Which of the following is true.

A. Area Square < Area hexagon < Area Circle

B. Area Square > Area hexagon > Area Circle

C. Area Square = Area hexagon = Area Circle

D. Area Square > Area Circle > Area hexagon

E. It is impossible to tell from the information given