AREA AND PERIMETER:AREAS OF REGULAR POLYGONS

Keystone Geometry

Review: Inscribed & Circumscribed with Polygons and CirclesInscribed means written insideCircumscribed means written around (the outside)

Def: A polygon is inscribed in a circle & the circle is circumscribed about the polygon when each vertex of the polygon lies on the circle.

inscribed polygoncircumscribed circle

Def: A regular polygon is a polygon that is equiangular & equilateral.

Inscribed Regular Polygons & Triangles

Total of Interior Angles = 540Each Interior Angle = 108

Inscribed Regular Pentagon

5 congruent isosceles triangles

Total of Central Angles = 360Each central angle = 72

Parts of a Regular Polygon

• A stands for AreaA(nonagon) is the area of a regular 9-sided figure.

• n is the number of sides of a regular polygon• p is perimeter, r is radius, s is side• a is apothem

• Apothem – The line segment from the center of a regular polygon to the midpoint of a side or the length of this segment.

• Sometimes known as the inradius, or the radius of a regular polygon’s inscribed circle.

Regular Polygon Area Theorem

Regular Polygon Area Theorem: The area of a regular polygon is one half the product of the apothem & the perimeter.

A(n-gon) = ( )nA XOYn

12sa

12a(ns)

12ap where, p = the perimeterYX s

O

a

Given: an inscribed regular n-gon (shown as an octagon)

12ap

Regular Polygon Terminology

Center of a regular polygon - the center of the circumscribed circle (O).

Radius of a regular polygon - the distance from the center to a vertex (OX).

Apothem of a regular polygon - the (perpendicular) distance from the center of the polygon to a side. (OM)

Central angle of a regular polygon - an angle formed by 2 radii drawn to consecutive vertices. ( )XOY

YX M

O (Regular Octagon)

Example: Equilateral (regular) Triangle

A12ap

r 2(4)

hyp2short

r 2a

pns

p3[(2 3(4)]

12

4(24 3)

r 8x 3(4)

p3(2x)

long 3 short

p24 3

30

ra

s

a = 4. Find r, p, A .

x 48 3

Example: Square (regular Quadrilateral)

A12ap

a8

hypleg 2

8 2 a 2

pns

p4(2x)

p4[(2)(8)]

p64

12

8(64)

256

45x a8

r a

r = . Find a, p, A.8 2

s

x

Example: Regular Hexagon

A12ap

x 5

long 3 short

a 3 x

pns

6(2)(5) 12

(5 3)(60)

hyp2 short

r 2(5)

6(2x)

6060

r 10

5 3 3 x150 3

s

a = . Find r, p, A.5 3

r a

x

Regular Nonagon

A12ap

a9.397

sin X opphyp

sin 70 a

10

pns

cosX adjhyp

12

(9.397)(61.56)9(2x)

70

r = 10; Find a, p, A.

r a

x

sX

Examples

r a A

1.

2. 8 5

3. 49

4.

r a p A

5. 6

6. 4

7. 12

8.

ra

ra

More Examples

ra

ra

1. r = , find A.2. a = 6, find A.

3. a = 8, find p.4. r = 12, find s.

5. s = 8, find r.

x x

ra

s

x