Areas of Polygons & Circle

Geometry Lesson 2 (2nd OP)

To every polygonal region, there corresponds a unique positive number called the area of the region.

Area of the polygonal region refers to the number of square units that exactly fill it.

Area Postulate

1 2

5 6

9 10

13 14

3 4

7 811 12

15 16

1 2

3 4

1 2 3

4 5 6

7 8 9

1 2

2

31

3

4

4

1

FACTS: All sides are equal. All angles are equal to 90 degrees.

SQUARE

s

s

A = s2

If s = 5 units If s = 1 /2 inches

If s = 3.5 units If s = 2/3 inches

If s = 7 inches If s = √5 dm

If s = 15 cm If s = (x+2)

A = s2

FACTS: 2 parallel sides Length and width All angles are equal to 90 degrees

RECTANGLE

L

Lw

w A =Lw

l= 5 units ; w= 2 units l= 3.5 units ; w= 2.2 units I= 7 in ; w= 4 in l= 15 cm ; w= 3 2/3 cm l=1/2 m ; w= 1/3 m l=√25 dm ; w= √4 dm l=√10 dm ; w= √5 dm l=2√x in ; w= √x in I= (x+2) ; w= (x+1)

A =Lw

FACTS: 2 parallel sides Length/base and altitude/height (width of the

rectangle)

PARALLELOGRAM

b

haA =bwh

l= 15 cm ; w= 3 2/3 cm l=1/2 m ; w= 1/3 m l=√16 dm ; w= √2.5 dm l=2√3 dm ; w= √3 dm l=2√x in ; w= √2x in I= (x+5) ; w= (x-2)

A =Lh

TRIANGLE

L(base)

h

L(base)

h

A =bh A =bh

A =bh 2

FIND THE AREA OF THE TRIANGLE: b=4; h=6

b=12 ; h=3

b=(2x+5) cm; h= 5 cm

b=(4x-5) in;h=(4x+5) in

A =bh 2

If height(a) or base(b) is unknown(x), given the third side (hypotenuse).

A =bh 2

[only for right Δs]

•Pythagorean Theorem c2 = a2 + b2

•Trigonometric Functions SOH, CAH, TOA CHO, SHA, CAO

a

45-45-90 special right Δ

b

45

45

h

(a=b)

(b=a)

(n√2)

•If a=1, b=____, h=____; A=•If b=2, a=____, h=____; A=•If a=3, b=____, h=____; A=•If h=10√2, a=____,b=____A=____•If h= 23√2, b=___,a=____A=_____

(n)

(n)

a

30-60-90 special right Δ

b

30

60

h

•If a=√3, b=____, h=____; A=_____

•If b=2, a=____, h=____ ; A=_____

•If a=3√3, b=____, h=____; A=_____

•If h=10,a=____, b=____; A=_____

•If h= 20,b=____, • a=____; A=_____

(2n)

(n)

(n√3)

Equilateral Triangle

FACTS

All sides are equal All angles are equal –

60 degrees

AEq=s2√3 4

If s=3 , A=_____ If s=2.5 cm, A= _____ If s=12, A=______ If s=200 ½ mm, A= _____ If s=11√4 m, A= _____ If s=4x in, A= ______ If s=(2x+2) m, A= _____ If s=(6x3-4y) in, A= ______

AEq=s2√3 4

FACTS: 2 bases (b1 and b2) that are parallel. Leg/s height

TRAPEZOID

BASE

LEG

LEG

h

BASE

A =bh 2

A =(b1+b2)h

2

If b1=5 cm, b2= 10 cm, h=5 cm, A=_______ If b1= 4x, b2= 12x, h=2x, A=_________ If b1=(2x+1) in, b2=(x-1) in, h=10 in,

A=___

A =(b1+b2)h 2

FACTS: 4-sided polygon with all sides equal. No angle is equal to 90 degrees.

RHOMBUS

d2

d1

d2

d1

A=1/2 d1d2

If a=3, b=4, A=____ If a=4 m, b=8, A= _____ If a=(2z) in, b=(4z) in, A=______ If a=6 in, c=12 in, A=______ If a=2, c=6 ,A=______ If b=4, c=12, A=______ If b=8, c=24, A=_______

A=1/2 d1d2

a

b

c

From parallelogram: A = bh = ½ C (r) = ½ (2Πr)(r)To circle: A= Πr2

Area of a CIRCLE

Aʘ=Πr2

r

r=2.5 cm, Aʘ=_______ r=√5 cm, Aʘ=_______ r=7√2cm, Aʘ=_______ d=16 mm, Aʘ=_______ d=18 in, Aʘ=_______ d=20√4 cm, Aʘ=_______

Aʘ=Πr2