2 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.
3 Sum of interior anglesDraw all the diagonals from one vertex. This will divide the polygon into triangles.
4 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
9 11.2 Areas of Regular Polygons Regular Polygon: all sides are same lengthYou 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)
10 ExampleFind the area of an equilateral triangle with 8-inch sides.
11 VocabularyWhen dealing with a polygon, think of it as if it were inscribed in a circle:
12 VocabularyCenter of a polygon: the same as the center of the circumscribed circleRadius of the polygon: the same as the radius of the circumscribed circleG is the center of the polygonGA is the radiusFAGEBDC
13 VocabularyApothem of the polygon: the distance from the center to any side of the polygon.The apothem is the segment GH.FAHGEBDC
14 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 =64
15 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.
16 Examples Find the area of the regular octagon. P = __________ Apothem = ___________Area = _____8.34.3
17 11.3 Similar FiguresIf 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
18 Similar FiguresThe 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.
19 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 – degreesLength of an arc – linear unitsThe circumference C of a circle is:.d is the diameter of the circler is the radius of the circle
20 Arc Length CorollaryThe 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 =APB
21 Arc Lengths The length of a semicircle = ½ of the circumference. The length of a 90 arc = ¼ of the circumference.
34 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.
35 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 CDLength of ABACDB
36 Probability and AreaLet 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 MArea of JJM
37 ExamplesFind 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.RTUS
38 ExamplesFind the probability that a randomly chosen point in the figure lies in the shaded region.
39 ExamplesFind the probability that a randomly chosen point in the figure lies in the shaded region.