# Area of Polygons and Circles

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Area of Polygons and Circles
Chapter 11

11.1 Angle Measures in Polygons
The sum of the measures of the interior angles of a polygon depends on the number of sides. Determining how many triangles are in each polygon will help you figure out the sum of the measures of the interior angles.

Sum of interior angles Draw all the diagonals from one vertex. This will divide the polygon into triangles.

Polygon Interior Angle Theorem
The sum of the measures of the interior angles of a convex n-gon is (n-2)*180 . The measure of each interior angle of a regular n-gon is

Examples Find the value of x.

Polygon Exterior Angle Theorem
The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 . The measure of each exterior angle of a regular n-gon is

Example Find the value of x.

Example Find the value of x.

11.2 Areas of Regular Polygons
Regular Polygon: all sides are same length You know that the area of a triangle is equal to A = ½ bh. If you are dealing with an equilateral triangle there is a special formula: A = (s = side)

Example Find the area of an equilateral triangle with 8-inch sides.

Vocabulary When dealing with a polygon, think of it as if it were inscribed in a circle:

Vocabulary Center of a polygon: the same as the center of the circumscribed circle Radius of the polygon: the same as the radius of the circumscribed circle G is the center of the polygon GA is the radius F A G E B D C

Vocabulary Apothem of the polygon: the distance from the center to any side of the polygon. The apothem is the segment GH. F A H G E B D C

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 = 6 4

Central angle of a regular polygon
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 (n) to find the measure of each central angle.

Examples Find the area of the regular octagon. P = __________
Apothem = ___________ Area = _____ 8.3 4.3

11.3 Similar Figures If two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their areas is a2:b2

Similar Figures The ratio of the lengths of corresponding sides is 1:2. The ratio of the perimeters is also 1:2. The ratio of the areas is 1:4.

11.4 Circumference and Arc Length
Circumference of a circle: the distance around the circle. Arc length: a portion of the circumference of a circle. Measure of an arc – degrees Length of an arc – linear units The circumference C of a circle is: . d is the diameter of the circle r is the radius of the circle

Arc Length Corollary The ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360 . Arc length of AB = A P B

Arc Lengths The length of a semicircle = ½ of the circumference.
The length of a 90 arc = ¼ of the circumference.

Examples Find the length of the arc.

Examples Find the length of the arc.

Example Find the circumference.

Examples Find the measure of XY.

11.5 Areas of Circles and Sectors
Area of a Circle = Find the area of the circle. 8 in.

Examples Find the diameter of the circle if the area is 96 cm2. Z

Sector of a Circle Sector of a Circle: the region bounded by two radii of the circle and their intercepted arc.

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 360 . .

Examples Find the area of the sector.

Examples A and B are two points on the circle with radius 9 inches and m APB = Find the areas of each sector.

Finding areas of Regions
= Area of shaded region Area of circle Area of polygon

Examples Find the area of the shaded region.

11.6 Geometric Probability
Probability is a number from 0 to 1 that represents the chance that an event will occur. Geometric Probability is a probability that involves a geometric measure such as length or area.

Probability and Length
Let AB be a segment that contains the segment CD. If a point K on AB is chosen at random, then the probability that it is on CD is: P(Point K is on CD) = Length of CD Length of AB A C D 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: P(Point K is in region M) = Area of M Area of J J M

Examples Find the probability that a point chosen at random on RS is on TU. Find the probability that a point chosen at random on RS is on TU. R T U S

Examples Find the probability that a randomly chosen point in the figure lies in the shaded region.

Examples Find the probability that a randomly chosen point in the figure lies in the shaded region.