Tuesday, August 7, 2007

Geometry Chapter 11 Area of Polygons and Circles

Chapter 11.1 Angle Measures in Polygons:

I) lets look at the sum of polygon angle measures:
Polygon - Number of sides - Number of triangles can make from the polygon - sum of the measures of the interior angles:
A) Triangle - 3 sides - 1 triangle - 180°
B) Quadrilateral - 4 sides - 2 triangles - 360°
C) Pentagon - 5 sides - 3 triangles - 540°
D) Hexagon - 6 sides - 4 triangles - 720°
E) Heptagon - 7 sides - 5 triangles - 900°
F) Octagon - 8 sides - 6 triangles - 1080°
G) Nonagon - 9 sides - 7 triangles - 1260°
H) Decagon - 10 sides - 8 triangles - 1440°
I) Undecagon - 11 sides - 9 triangles - 1620°
J) Dodecagon - 12 sides - 10 triangles - 1800°

Do you see a pattern? The number of sides minus 2 is equal to the number of triangles so...

II) Theorem:
A) Polygon Interior Angles Theorem - the sum of the measures of the interior angles of a convex n-gon is (n - 2) (180°)
Example: What is the sum of the measures of the interior angles of a convex 20-gon?
(20 - 2)(180°) = 3240°
B) The measures of each interior angle of a regular n-gon is:
(1/n)(n-2)(180°)
Example: Given a regular hexagon, what is the measure of each angle?
(1/6)(6-2)(180°) = (1/6)(4)(180°) = 120°
C) Polygon Exterior Angles Theorem - the sum of the measures of the exterior angles of a convex polygon, one angle iat each vertex, is 360°.
Example: If the exterior angles of a polygon is x, 2x, 2x, 4x, and 3x, solve for x.
x + 2x + 2x + 4x + 3x = 360
12 x = 360
x = 30
Example: Given a regular polygon has 20 sides, what is the measure of an exterior angle and an interior angle?
360/20 = 18, so every exterior angle is 18°
180° - 18° = 162° , so every interior angle is 162°

(20 - 2)(180) = (162)(20)
3240 = 3240 so it checks

Example: Given the measrue of each interior angle of a regular n-gon is 120°, how many sides does the polygon have?
(1/n)(n-2)(180) = 120
(n-2)(180) = 120n
180n - 360 = 120n
60n = 360
n = 6

If the interior angle is 120°, then the exterior angle is 60°
360/60 = 6 sides, same answer!

Chapter 11.2 Areas of Regular Polygons:

I) Area of an Equilateral Triangles is:
A = ((√3)/4) s2
Example: Given an equilateral triangle with side length of 4 what is the area?
A = ((√3)/4)(42)
A = ((√3)/4)(16)
A = 4√3 square units

II) Vocabulary:
A) Center of the polygon - when you circumscribe a circle about a regular polygon, the center point of the circle is the center of the polygon.
B) Radius of the polygon - when you circumscribe a circle about a regular polygon, from the center point of the circle to the vertex of the polygon is the radius of the polygon.
C) Apothem of the polygon - the distance from the center point of the polygon perpendicular to any side of the polygon.
D) A Central angle of a regular polygon - is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. You can divide 360° by the number of sides to find the measure of each central angle of the polygon.

III) Theorem:
A) Area of a Regular Polygon - the area of a regular n-gon with side length "s" is half the product of the apothem "a" and the perimeter "P"
A = (1/2)(a)(P)

Chapter 11.3 Perimeters and Areas of Similar Figures
I) Theorems:

A) Areas of Similar Polygons- if two polygons are similar with the lengths of the corresponding sides in the ratio of a:b, then the ratio of their areas is a2:b2
Example: Given two similar hexagons with corresponding sides of 4 and 5, what is the ratio of their areas?
42:52
16:25

Chapter 11.4 Circumference and Arc Length
A) Circumference
- of a circle is the distance around the circle.
C = 2π r = πd
B) Arc length corollary:
in a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360º
Example: recall, the central angle is equal to it's intercepted arc
Given circle P with central angle APB = 30º and radius of 8 radius, what is the measure of the intercepted arc AB?
(measure of the arc AB)/(2π8) = (30/360)
(measure of the arc AB)/ (16π) = (1/6)
measure of the arc AB = 16π/6
measure of the arc AB = 8π/3 inches or about 8.37758041 inches

Chapter 11.5 Areas of Circles and Sectors
I) Vocabulary:
A) Sector of a Circle - is the region bounded by two radii of the circle and their intercepted arec.

II) Theorems:
A) Area of a Circle - the are of a circle is π times the square of the radius
A = πr2
Example: given a circle with radius of 9 cm, what is the area?
A = π(9)2 = 81π square cm.
B) Area of a Sector - the ratio of the area A of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360º
Example: Given Circle C with points A and B on the circle, the major arc AB is 293º and the radius is 10 cm, what is the area of the sector?
(Area of sector)/(π 102) = 293º /360º
Area of sector/100π = 293/360
Area of sector = 1465π/18 which is about 255.6907354 square cm.

Chapter 11.6 Geometric Probability
I) Vocabulary
A) Probability is a number from 0 to 1 that represents the chance that an event will occur.
B) Geometric Probability - involves geometric measures such as length or area

II) Theorems:
A) Probability and length - Let line segment AB be a segment that contains the segment CD. If a point K on line segment AB is chosen at random, then the probability that it is on CD is as follows:
P(Point K is on line segment CD) = (length of CD)/(length of AB)
Example: Given AB = 5 and CD = 2, then the probability that point K is on CD is
5/2

B) Probability and Area - Let J be a region that contains region M. If a point K in J is chosen at random, then the probability that it is in region M is as follows:
P(Point K is in region M) = (Area of M)/(Area of J)
Example: You mow a rectangular lawn of 20 feet by 30 feet and it has a garden of 5 feet by 6 feet. What is the geometric probability that if someone tosses a ball that it will land in the garden?
(5)(6) = 30 square feet
(20)(30) = 600 square feet so
30/600 = 1/20