Download Areas of Circles and Regular Polygons

Warm-Up
1. Find the perimeter
of the polygon:
160 m
2. Find the area of the green shaded region:
7cm
4 cm
2 cm
24 cm²
Math Jokes of the Day!!!
Which triangles are the coldest?
Ice-sosceles triangles.
What did the student say
when the witch doctor
removed his curse?
Hex-a-gon!
SECTION 11.3
(PART 1)
AREAS OF CIRCLES AND
REGULAR POLYGONS
5/1/14
Learning Goals for Day 1 of 11.3

You will all be able to:
 Define
a polygon and use prefixes that describe
polygons according to their number of sides
 Compute the area of a circle
 Define our new vocabulary word of the day!
 Use what you know about perimeters and common area
formulas to compute the area of a Regular Polygon
 Compute areas of shaded regions within another
polygon or circle
What is a polygon?




Polygon – The word "polygon" derives from the Greek
words poly (many) and gonu (knee). So a polygon is a thing with
many knees! It must have straight edges that connect at vertices.
Polygons can be concave or convex and irregular or regular.
“Regular” – all sides congruent
Today we will just be
focusing on REGULAR
Polygons (Specifically
equilateral triangles, squares,
& hexagons)
Polygon Prefixes…
Prefix
Meaning (# of sides)
Tri -
3
Quad -
4
Penta -
5
Hexa -
6
Hepta -
7
Octa(o) -
8
Nona -
9
Deca -
10
Poly -
Many
Fun Fact:
The more sides a
polygon has, the
more “circular” it
becomes!
Area of a Circle
Area of ⊙ = π times the (radius)²
Example #1:
Find the exact Area of
⊙ C , which has a
diameter of 14 cm.
 C
A ⊙ = πr²
A ⊙ = π(7cm)²
A ⊙ = 49πcm²
Don’t forget your
units and that they
are squared,
because it’s an area
problem!!!
Let’s Break Down a Regular Polygon…

A regular polygon with “n” sides is comprised of “n”
congruent, isosceles triangles.

The base of each triangle is one side of the
polygon.

The two congruent legs of each isosceles triangle are
considered the radii of the polygon. The radii start
from the center point and end
at each vertex.

The height of each triangle is called an apothem.
Central Angles of Regular Polygons
The central angles of a regular polygon are all
congruent and add up to 360° (Central ∡ Sum Theorem ⊙)
 To find the measure of one central angle, use the
following formula:

One Central ∡ = 360° ÷ n
“n” being the number of sides the
polygon has
Example #2:

Find the central angle for each regular polygon:
A.
B.
n = 10
n=6
Central ∡ = 360° ÷ 10
Central ∡ = 360° ÷ 6
Central ∡ = 36°
Central ∡ = 60°
New Vocabulary Word of the Day!

Apothem – a line segment drawn from the center
of a polygon perpendicular to one of its sides,
bisecting that side and the central angle.

*** It is a PERPENDICULAR BISECTOR! ***
Sounds like …. A Opossum
Why is the Apothem Important?


The apothem is needed when finding the area of a regular polygon.
(It’s the height of each isos. △ in the polygon)
The formula for the area of a regular polygon is:
A = ½ the apothem times the perimeter
or
You can think of area of the entire polygon as
ADDING
_____________________
up the area of all of the
ISOSCELES
TRIANGLES that
___________________
________________
make up the polygon; hence using the PERIMETER!
The perimeter is just the base of each triangle added
together!!!
When finding the Area of a Regular
Polygon….

You might be given just the perimeter, or one side
length, or the apothem, or the radius… Or you
may be given a combination of these facts.
 (Best
case scenario is where you are given the
perimeter/or side length, and the apothem because
you are given everything for the Area formula!)

Depending on what you are given, just reminder
that you always have the central angles to rely on
and the fact that the apothem bisects this angle
and the side length.
Quick Refresher of Special △’s
30°-60°-90°
Hypot = twice (short) or 2(short)
Long Leg = (short)
45°-45°-90°
Hypot = leg
Example #3:

Find the area of the following regular polygon with a
radius of 4 cm.
** What do we still need in
order to find the area?
A = ½ a·P
Central ∡= 360°÷3
= 120°
One side length = 4 cm
P = 3(4
cm) = 12
cm
A = ½ a·P
A = ½ (2cm)·(12
A = 12
cm²
a
60°
4cm
2 cm
30°
Apothem bisects each central angle,
so 120° ÷ 2 = 60° each
2
cm
cm)
Example #4:

Find the area of the following regular polygon:
A = ½ a·P
A = ½ a·60cm
60°
60°
60°
60°
60°
A=½5
cm·60cm
A = 150
cm²
30°
60°
30°
5
60°
5cm
cm
Example #5:

A square is circumscribed about a circle. Find the area of the shaded
region if the circle has a radius of 8m:
Area of Shaded Region = Area  - Area 
Area  = (16m)(16m) = 256 m²
Area  = π·(8m)² = 64π m²
8m
16m
Area = (256 – 64π) m²
SECTION 11.3
(PART 2)
AREAS OF CIRCLES AND
REGULAR POLYGONS
5/2/14
Right Triangle Trigonometry!
SOH – CAH - TOA
Ex 1: Find the area of the regular nonagon with a perimeter
of 180 cm .
Ex 2: Find the area of the regular heptagon with an apothem
of 6 in.
Ex 3: Find the area of the non-shaded region if the circle has
a radius of 17 ft .