Areas of Regular Polygons and Circles

Objective: TLW find areas of regular polygons and circles. SOL G.14b

Do- Now 04-14-2014

Warm UP

Go over Test

Review beginning slides from before break

TLW find areas of regular polygons and circles.

TEST FRIDAY!!

No Homework

Find the area of each Figure Below:

6.79 cm2 214.16 yd2

54 u2

RECALL:

What is the definition of a regular polygon? A Regular Polygon is a convex polygon in which all

angles are congruent and all the sides are congruent Context of the definition; What does convex refer to?

Every internal angle is less than or equal to 180o Every line segment between two vertices is in or on the

polygon

How could this information assist us in finding the area of a regular polygon?

Looking at a Regular Hexagon:

Look at the Hexagon given below:

Consider the following questions:

What could we do to hexagon ABCDEF in order to find the height of h ? Inscribe the polygon in a circle How does this help us?

We can now evaluate the area using Radius

What would be our radii? GE & GF How does this help us?

We can now use SOHCAHTOA to find lengths and angle measures

h

Looking at a Regular Hexagon:

Using the Hexagon given below: Looking at the image, line segment

GH is drawn: from the center of the regular

polygon perpendicular to a side of the polygon.

This perpendicular segment, or height, is called the apothem. (Labeled by the “h”) How does labeling this

segment help us find the area of the regular hexagon?

What do we know about the central angles of a regular polygon? Does this help us? If so,

how?

hh

Looking at a Regular Hexagon:

Using the Hexagon given below:

By labeling and defining h as our perpendicular bisector, we know that ΔEGF is an Isosceles Triangle. We know that both of the radii are congruent in our triangle, that h bisects angle EGF making those angles congruent, and they share congruent right angles.

h

What would happen if we drew line segments from the center t each vertex on the hexagon?

we end up with 6 congruent Δ’s

Wrapping Up: Regular Hexagons

How can we use this information to determine the area of our hexagon in square units?

If we have 6 repeating triangles, how can we used this information to find the area of a hexagon? Remember the area of a

rhombus

h

Rhombus: (1/2)bh = [(1/2)((1/2)d1 )((1/2)d2)]4

= (1/2) d1d2

Hexagon: (let a = h) = [(1/2) ab] # of sides

= [(1/2) ab] 6= (1/2) 6ba what is 6b?

Area of a Regular Polygon:

If a regular polygon has an area of A square units, a perimeter of P units, and an apothem of a units, then:

Let’s begin with an example using a regular pentagon: What is the Perimeter of MNOPQ when QP = 12

inches? P = # of sides side length = 5 (12 in.) = 60 inches

What is the Area?

= 247.80 in2

SOHCAHTOA

How do we find the apothem(a)?360/ 5 = 72o

72/ 2 = 36o

Tan (36o) = (6/a) a = (6/(tan(36o))) = 8.26 cm

A = (1/2)Pa

How can we use all of this information to find the area of a circle? Can we get to the area of

a circle from the equation for Perimeter?

Area of a Circle

Can we use the area of a

regular polygon?

Yes! A = (1/2) Pa PC = 2πr Plug in!

A = (1/2)(2πr)a What’s our “a”? It’s our

radius!! A = (1/2)(2πr)(r)

= πr2

If a circle has an area of A square units and a radius of r units, then:

A = πr2

Let’s Try One!

Let the circle shown below have a radius of 9 centimeters, what is the perimeter and the area of the circle?P = 2πr = 2π(9) = 18π cm ≈ 56.55 cm

A = πr2

= π(9)2

= (99)π cm2

= 81π cm2

≈ 254.47 cm2

Your Turn!!

What if we want to know the area of the shaded region around an inscribed

polygon?

Let r = 12.5 inches

What do we need to calculate??

Area of a Circle:= πr2

= π(12.5)2

= 156. 25π in2

≈ 490.87 in2

Area of the Square:= s2

s2 + s2 = (2r)2, where r = 12.5 2s2 = (25)2 = 625 s2 = 312.5 in2

Square root both sides! s = 17.68 inches

Area of a Circle – Area of a Square:(490.87 in2) – (312.5 in2)

178.37 in2