GEOMETRY – AREA UNIT
Day I – Area Review
Objectives: SWBAT find the area of various shapes
Triangle Parallelogram Rectangle
Square Trapezoid Kite
1b
2b
1d
2d
triangle
1A base height
2
1A b h
2
parallelog ramA base height
A b h
rectan gleA base height
A b h
2
square
2
square
A side
A s
trapezoid 1 2
trapezoid 1 2
1A height base base
2
1A h b b
2
Kite 1 2
Kite 1 2
1A diagonal diagonal
2
1A d d
2
Rhombus Circle Sector
Find the area of the following shapes.
1. 2. 3.
r hombus 1 2
r hombus 1 2
1A diagonal diagonal
2
1A d d
2
2
circle
2
circle
A radius
A r
2
sector
2
sector
CentralA radius
360
NA r
360
rectan gle
2
A b h
15 5
75 ft
2
square
2
2
A s
3
9 in
triangle
2
1A b h
2
110 4
2
20 units
4. 5. 6.
trapezoid 1 2
2 2 2
2
2
trapezoid 1 2
trapezoid
trapezoid
trapezoid
2
trapezoid
1A h b b
2
Need Height
Use Pythags
3 h 5
9 h 25
h 16
h 4
1A h b b
2
1A 4 10 16
2
1A 4 26
2
A 2 26
A 52 in
parallelog ram
parallelog ram
2
parallelog ram
A b h
A 8 15
A 120 cm
r hombus 1 2
1
2
r hombus
r hombus
2
r hombus
1A d d
2
d 2 2 4
d 10 2 12
1A 4 12
2
A 2 12
A 24 units
7. 8. 9.
50
trapezoid 1 2
22 2
2
2
trapezoid 1 2
trapezoid
trapezoid
trapezoid
2
trapezoid
1A h b b
2
Need Height
Use Pythags
1 h 50
1 h 50
h 49
h 7
1A h b b
2
1A 7 11 12
2
1A 7 31
2
A 3.5 31
A 108.5 cm
10 2
5
Hypo Small
b
5 3
5 3
Small Middle / Long
h
triangle
2 2
1A b h
2
18 5 5 3
2
113 5 3
2
65 3mm or 56.29 mm
2
728
28 7
1
28 1 7
0 532 7
13 165
Oppositetan C
Adjacent
tanb
tan
b
Cross Multiply
b tan
b .
b .
rectan gle
2
A b h
13.165 7
92.156 in
10. 11. 12.
3 2 cm
3 2 2
3
Hypo Leg
s
2
square
2
2
A s
3
9 in
2
sector
2
sector
sector
sector
2
sector
NA r
360
90A 14
360
90A 196
360
1A 196
4
A 49 mm
2
sector
2
sector
sector
sector
2
sector
NA r
360
120A 7
360
120A 49
360
1A 49
3
49A cm
3
Day II – Area of Shaded Figures
Objectives: SWBAT find the area of various shapes and shaded regions
Area of a Shaded Figure: If you know the shape of the shaded area – just use the formula
If it isn’t a shape you recognize then think of it as BIG SHAPE – LITTLE SHAPE
Space Big Small
Inbetween Shape Shape
Examples: Find the area of the shaded area.
1. 2.
2
Rectangle-Rectangle
7 15 4 15
10
45
5 60
Space Big Small
Inbetween Shape Shape
Shaded
Area
Shadedh b h b
Area
Shaded
Area
Shaded
Ar
Shadedin
Area
ea
2
2
2
278
-Rec
tangle
10 9 4
314 36
Space Big Small
Inbetween Shape Shape
ShadedCircle
Area
Shadedr h b
Area
Shaded
Area
Shaded
Area
Shadedin
Area
3. 4.
5 2 cm
2 2
2
22
-Circle
16 8
256 200.
55.0
6
4
9
Space Big Small
Inbetween Shape Shape
ShadedSquare
Area
Shadeds r
Area
Sha
Sh
ded
Area
Sha
adedin
A
ded
Area
rea
2
16 2
8
Diameter radius
r
r
22
1
2
2
2
2
2
5 5 2
2
1
2
15 2 7
2
15 9
22.5
2
2.5 9
5 50
25
5
Need the Height
h
h
h
Trapezoid
A h b b
A
A
A
Shadedcm
h
Area
5. 6.
2
2
2
Square 4Rectangles
4
1
8
3 4 4 5
169 8
9
0
Space Big Small
Inbetween Shape Shape
Shaded
Area
Shadeds h b
Area
Shaded
Area
Shaded
Are
Shadedcm
Are
a
a
2
Rectangle 2Triangles
12
46
2
32 16 2 4 6
512 48
4
Space Big Small
Inbetween Shape Shape
Shaded
Area
Shadedh b h b
Area
Shaded
Area
Shaded
Area
Shadedft
Area
Day III – Area of Compound Figures
Objectives: SWBAT find the area of various shapes
Area of a Compound Figures or Unfamiliar Shapes:
Break it up into shapes that you know then ADD UP THE SHAPES
ONLY DRAW Straight Lines
Examples: Find the area of the given shapes.
1. 2.
2
Re
44
ctanRecta gle
8 4
3
ngle
2 6
12 2
h b
Total
Total
Total
Total
Total
b
m
h
c
2
Re
45
ctanRecta gle
5 8
4
ngle
3 5
15 0
h b
Total
Total
Total
Total
Total
b
t
h
f
3. 4.
2
TrianglRectangle
11 18
2
e
1
2
16 8
2
24198
22
Total
Total
Tot
h
al
Total
h
Total c
b b
m
2 2 2
2
2
Triangle
8 10
64 100
36
6
Need Base
b
b
b
b
1 2
2
Trapazoid
1
2
1
94
Rectangle
1 814
14
14 62
8
0
Total
Total
T
h
otal
Total
b
Total cm
h bb
5. 6.
1
2
2
+Rectangle
3 15
45
Triangle
17 9
153
Trapezoid
1
2
1
22
4 4 92
26
4
h
Total
Total
Total
Tota
h b bh b b
l
Total in
2
10 2
5
Diameter radius
r
r
2
1
2
2
2
Rectangle
12 1
Tr
222.
apezoid +Half Circle
1
2
15
2
39.25
0
1
1
2
19 4 10
2
63
25
20
Total
Total
T
h
otal
Total
Total cm
h rb b b
Day IV – Area of Regular Polygons – Day 1
Objectives: SWBAT find the area of regular polygons
Regular Polygons: Polygons with congruent sides and angles
Apothem:
A segment from the center that is a Perpendicular Bisector of a side
Radius: A segment from the center that bisects An angle of the polygon
Perimeter: Add up all the side or multiply the number of sides and the length
Area of a Regular Polygon Formula:
1
2
1
2
A apothem Perimeter
A aP
Examples: Find the area of the following regular polygons
1. 2. 3.
2
9 5 4
1
2
15 45
112.5
5
2
A aP
A
Perime
A n
t
i
er
2
2.0 8 1
1
2
12.4 16
2
19.2
6
A
Perimete
i
r
A
aP
A
n
2
8 6 4
1
2
15.5
132
8
482
A aP
A
Perimete
A
r
ft
4. 5.
6.
3 3 ft
2
4
16
1
2
12 16
2
4 16
A
Perimeter
P
A
A in
a
2 2 2
2
2
3 6
9 36
27
3 3
Need Apothem
a
a
a
a
2
2
1
2
13 3
6
9 3
46.76
3 18
1
5
82
A in
or
A aP
A i
Per m
n
e
A
i ter
2
2
1
2
13 3
54
6 6
3
93.531
36
362
A
A in
a
o
P
A
r
A
Perimeter
in
3 3
6
Need Side
Given Half Side
Side
Side
Day V – Area of Regular Polygons – Day 2
Objectives: SWBAT find the area of regular polygons using Special Right Triangles
Area of a Regular Polygon Formula:
1
2A apothem Perimeter
1
2A aP
Central Angle: angle made by two radii Radius: A segment from the center that bisects An angle of the polygon
Side:
Half Side: Half of the side of a pentagon Steps for Finding the Area of a Regular Polygon 1 – Find Central Angle
2 – Divide the Central Angle in Half to find the top angle in your triangle
3 - Ask yourself what you have and label your Triangle Do I have a radius/hypo?
Do I have the apothem? Do I have a side? Whatever it is, label your triangle!
4 – Find out what you need to complete the formula.
If you need the apothem or “a”, use SOH-CAH-TOA / 2 2 2a b c
If you need side or “x”, use SOH-CAH-TOA / 2 2 2a b c
5 – Find the Perimeter Once you have “x,” double it and multiply that by the number of sides
6 – Input all data into the formula 1
2A aP
Radius Hypotenuseapothem
1
2x side
in Triangle2
CentralTop
360Central
n
Examples: Find the area of the following regular polygons
1.
2.
360
360
6
60
Centraln
Central
Central
4
8
Need Apothem
Half Side
Side
4tan 30
tan 30 4
1
6.
tan 30
92
4
4
ta
8
n 30
Cross Mulitp
a
a
a
ly
a
a
2
6 8 4
1
2
166.28
16.9
8
28 4
82
A aP
A
Perimet
A i
er
n
360
360
6
60
Centraln
Central
Central
6
Need Half Side
Apothem
ta
6 tan 30
6 tan
n 306
tan 30
1 6
3.464
6.928
30
Cross Mulitpl
x
x
x
Sid
y
x
x
e
2
6 6.928 41.5
1
2
16 41.568
2
124.7 4
68
0
Perimeter
A aP
A
A in
3.
4.
360
360
6
60
Centraln
Central
Central
12
Need Half Side
Need Apothem
Radius
sin 3012
sin 30
1 1
2
6
12 sin 30
12 s 0
1
i
2
n 3
Cross Mulitp
x
x
x
l
i
y
x
S
x
de
2
12 6 72
1
2
110.392 72
2
374.123
Perimete
A a
r
P
A m
A
m
12 cos 30
1
cos 3012
cos 30
1 12
10.392
2 cos 30
Cross Mulit
a
a
a
ply
a
a
360
360
6
60
Centraln
Central
Central
12
6
Need Apothem
Side
Half Side
tan 3
6tan 30
tan 30 6
1
10.392
0 6
6
tan 30
Cross Mulitply
a
a
a
a
a
2
6 12 72
1
2
110.392 72
2
374.112
Perimete
A ft
A aP
A
r
5.
360
360
3
120
Centraln
Central
Central
6
Need Half Side
Apothem
ta
2 tan 60
2 tan
n 602
tan 60
1 2
3.464
6.928
60
Cross Mulitpl
x
x
x
Sid
y
x
x
e
2
3 6.928 10.3
1
2
12
10.3
92
10.3922
92
Perimeter
A aP
t
A
A f
Day VI– Area of Regular Polygons – Day 3
Objectives: SWBAT find the area of regular polygons given only 1 piece of data
Examples: Find the area of the following regular polygons
1.
3
Need Half Side
Need Apothem
Radius
sin 303
sin 30
1 3
1
3 sin 30
3 sin 0
.5
3
3
Cross Mulitpl
x
x
y
x
x
x
Side
2
1
2
3 6 18
23.383
12.598 18
2
A aP
Perimeter
A
A cm
cos 303
cos 30
1 3
2.
3 cos 30
3 co
598
s 30
Cross Mulitpl
a
a
a
y
a
a
360
360
6
60
Centraln
Central
Central
2.
3.
360
360
5
72
Centraln
Central
Central
2
Need Half Side
Apothem
ta
2 tan 36
2 tan 3
n 362
tan 362
1
1.453
2.9 6
6
0
Cross Mulitply
x
x
x
x
Side
2
5 2.906 41.568
1
2
12 2.90
2 9
62
. 06
A aP
A
A
Perime e
ft
t r
6
Need Half Side
Need Apothem
Radius
si
6 sin 22.5
6 sin 22
n 22.56
sin 22.5
1 6
2.296
4.
.5
592
Cross Mulitpl
x
x
x
Sid
y
x
x
e
2
8 4.592 36.736
1
2
15.5 3
101
6.
.819
43
2736
Perim
A
A f
aP
A
t
t
e er
360
360
8
45
Centraln
Central
Central
cos 22.56
cos 22.5
1 6
6 cos 22.5
6 cos 22.5
5.543
Cross Mulitpl
a
a
a
y
a
a
4.
360
360
8
45
Centraln
Central
Central
10
5
Need Apothem
Side
Half Side
tan 22.5 5
5
t
5tan 22.
an
5
tan 22.5 5
1
12.07
2.5
1
2
Cross Mulitply
a
a
a
a
a
2
10 8 80
1
2
112.071 80
2
482.842
Perimete
A a
r
P
A n
A
i
Day VII – Angles of Regular Polygons
Objectives: SWBAT find the area of regular polygons
Regular Polygons:
If it is equiangular and equilateral
All sides and angles are congruent
Sum of Internal Angles Formula of a Polygon
Find the value of x.
1. 2. 3.
2 180
Sum n
n Number of Sides
2 180
6
6 2 180
4 180
720
Sum n
n
Sum
2 8 7 3 6 4 720
6 720
120
x x x x x x
x
x
2 180
4
4 2 180
2 180
360
Sum n
n
Sum
3 90 90 360
4 180 36
4 180
45
x x
x x
x
x
2 142 2 3 14 3 14 540
10 170 540
10 370
37
x x x x
x
x
x
2 180
5
5 2 180
4 180
540
Sum n
n
Sum
Sum of Internal Angles Formula of a Regular Polygon
Given the following regular polygons, find the sum of the interior angles.
4. Hexagon 5. Nonagon 6. Octagon
You are given the measure of each interior angle of a regular n-gon. Find the number of sides.
7. 108 8. 144 9. 135
2 180
Sum n
n Number of Sides
2 180
9
9 2 180
7 180
1260
Sum n
n
Sum
2 180
6
6 2 180
4 180
720
Sum n
n
Sum
2 180
8
8 2 180
6 180
1080
Sum n
n
Sum
2 180
Re
nAn Angle
n
For gular Polygons Only
2 1801
2 180108
2 180108
1
108 2 180
108 180 360
180 180
72 360
5
nm
n
n
n
n
n
Cross Mulitply
n n
n n
n n
n
n
2 1801
2 180144
2 180144
1
144 2 180
144 180 360
180 180
36 360
10
nm
n
n
n
n
n
Cross Mulitply
n n
n n
n n
n
n
2 1801
2 180135
2 180135
1
135 2 180
135 180 360
180 180
45 360
8
nm
n
n
n
n
n
Cross Mulitply
n n
n n
n n
n
n
Sum of External Angles Formula of a Regular Polygon
External Angles add up 360 degrees
Solve for x.
10. 11.
Given the following regular polygons, find the number of degrees of
each exterior angle.
9. 6-gon 10. Pentagon 11. Heptagon
139 2 6 9 360
17 139 360
17 221
13
x x x
x
x
x
2 5 3 10 5 2 6 5 360
18 0 360
18 360
20
x x x x x
x
x
x
6 360
60
x
x
5 360
72
x
x
7 360
51.43
x
x
