Polygons

OBJECTIVES

• Exterior and interior angles• Area of polygons & circles• Geometric probability

PolygonsDefinition: closed figure/ coplanar segments/ sides

have common, non-collinear endpoints/ each segment intersects only at the endpoints

- most polygons will be convex: sides are ‘pushed out’— concave polygons have one vertex ‘pushed in’ the figure

- A ‘regular polygon’ has all sides & all 's

Interior & Exterior AnglesInterior (vertices) with n sides S = sum of

S = 180 (n –2)

Exterior of polygon with n sides

's 's

's Exterior angle Θ360

θ = n

n regular each interior exterior S = Sum of polygon angle angle θ interior angles3 Triangle 60° 120° 180°4 Quadrilateral 90° 90° 360°5 Pentagon 108° 72° 540°6 Hexagon 120° 80° 720°8 Octagon 135° 45° 1080°10 Decagon 144° 36° 1440°n n-gon 180(n-2)/ n 180-interior 180(n-2)

ParallelogramsDefinition: 2 pairs of parallel sidesany side can be a base for each base there is an altitude (or height)

AREA A = base • height = b h

Area of a complex region is the sum of its non-overlapping parts

h

Area of rhombi, triangles, & trapezoids • Congruent figures have equal areas

• Area of a triangleA = ½ b h

• Area of a trapezoid

A = ½ h (b1 + b2)

• Area of a rhombus

A = d1d2

h

base

h

b1

b2

d1 d2

Regular polygons and circlesAn apothem --center to side( | bisector)A radius of a polygon--center to vertexPerimeter of a polygon—sum of sides

Area regular polygon

A = ½ Perimeter • apothem = ½ Pacircles

A = π r2 radius

ra

Geometric probability =

Segment:: If C is between A & B, the probability of being on AC :

Polygon: Given random point C and rectangle A, the probability that C is in triangle B:

Circle: area of a sector = (N = central angle, r = radius) Probability of being in certain sector(s) =

successful area

total area

•A

•B

•Clength of AC

P = length of AB

area of BP =

area of A ABB

2N

360r

area of sector(s)P =

area of circle