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3-2-1 Formulas
Formulas appear in almost any profession. A formula is an expression where the variables and result
have a specific meaning. In the formula P=2w+2l the w and l are measurements and the result, P, is
the perimeter of a rectangle with those measurements.
Steps: Copy the pattern the formula makes.
Insert the known numbers for the variables.
Simplify.
Look through the examples. Then work the problems that follow. Be careful with the order of
operations as you simplify.
List of formulas
Geometry -- 2 Dimensional
Perimeter
Rectangle P=2w+2l
where w is width and l is
length
Example: P=2(3)+2(7)=20 in.
Square
7 in.
P=4s
where s is the length of a
side.
Example: P=4(5)=20 in
Trapezoid
3 in.
P=a+b+c+d where a,b,c
and d are the lengths of the
sides.
5 in.
10 in.
Triangle
P=a+b+c where a, b, and c
are the lengths of the sides of
a triangle.
Example: P=5+9+7=21 in
Circle
C=2πr where C is
circumference, r is the radius
and π is pi. (Use 3.14 for π)
Example: C=2π(3)=6 π
Approximately 18.85 ft
4 in.
5 in.
4 in.
Example:
P=5+10+4+5=24 in.
A=s2
where s is the length of a side.
Example: A=(5)2=25 in2
5 in.
5 in.
Area
A=lw
where w is width and l is
length
Example: A=(7)(3)=21 in2.
5 in.
7 in.
4 in.
9 in.
3 ft.
A= ½ h(b1+b2) where h is
the perpendicular height
between the bases and b1 and
b2 are the bases.
A= ½ (4)(4+10)=28 in2.
A= ½ (bh) where b is the
base and h is the height of the
triangle.
Example: A=½ (9)(4)=18 in2
A=πr2 where r is radius and π
is pi. (Use 3.14 for π.)
Example: A=π(3)2=9 π
Approximately 28.27 in2
1
Geometry -- 3 Dimensional
Surface Area
Rectangular
Solid
S=2lw+2wh+2lh
where l is the length,
w is width, and h is
height.
Volume
V=lwh where l is the
length, w is width, and h is
height.
4 m.
3 m.
Example:
V= (12)(3)(4)=144 m3
12 m.
Example:
S=2(12)(3)+2(3)(4)+
2(12)(4)=192 m2
Sphere
radius
3 mi.
SA= 4πr2 where r is
radius and π is pi.
Example: V=4/3π(3)3=36 π
=113.9mi3.
Example: SA=4π(3)2
=36 π113.9mi.
Cylinder
SA= 2πr2+ 2πrh
where r is radius, π is
pi, and h is the height
of the cylinder.
V=4/3πr3 where r is radius
and π is pi.
height of 6 cm and
radius of 4cm
V = πr2h where r is radius,
π is pi, and h is the height of
the cylinder.
Example:
V = π(4)2(6)=96 π301.59
cm3
Example:
SA=2π(4)2+ 2π(4)(6)
=80 π251.33 cm2
Cone
SA= πr2+πrs where r
is radius, π is pi, and s
is the length of the
slant of the cone.
Example:
SA= π(5)2+π(5)(13)
= 90π
282.74 m2
Pyramid
SA= 4(½ bh) + b2
where b is one side of
the square base and
h is the height of the
triangle face.
Example:
SA=4(½ (10)(13)) +
(10)2=360 m2
2
13 m
12 m.
V= 1/3 πr2h where r is
radius, π is pi, and h is the
height of the cone.
Example:
V= 1/3 π(5)2(12)
=100 π314.16 m3
5 m.
V=1/3 Bh where B is the
area of the base and h is
the height.
12 m.
13 m
10 m.
10 m.
Example: B=10x10=100
V=1/3 Bh=1/3 (100)(12)
=400 m3
Finance
Retail price
p=c+rc where p is
the price, c is the
wholesale cost and
r is the rate of
markup.
What is the retail cost of a sweater
with a wholesale price of $20 and a 75% markup?
p=20+.75(20)=$35
Sale price
p=c – rc where p
is the price, c is the
original cost and r
is the rate of
discount.
What is the sale price of a freezer
originally costing $450 on sale for 45% off?
p=450-(.45)450=$247.50
Simple Interest
I=Prt where I is
interest, P is the
principal, r is the
annual rate, and t
is the time in years.
Accumulated
amount with
compound
interest
r

A  P 1  
 m
Distance
mt
Where A is the final
amount after all the
interest is added.
r is the annual rate
as a decimal. m is
the number of
times the interest is
compounded a
year,
P is the principal,
and t is the number
of years.
A car is sold for $15000 with simple interest at 12% for a
period of 5 years. How much interest is paid? How much
total is paid back? What is the monthly payment?
I=Prt=(15000)(.12)(5)=$9000 (Write 12% as a decimal.)
Total = 15000+9000=$24000 (Add the principle to the
interest.)
Monthly Pmt. = 240000/60=$400
(Divide the total by the number of months.)
What is the accumulated amount if Delores deposits $5000
in her grandson’s account at a rate of 5% compounded
quarterly (4 times a year.) The money is in the account for
18 years.
 .05 
A  5000 1 

4 

F
Jack drove for 5 hours at 60 miles per hour. How far did he
go? d=60(5)=300 miles.
9C
 32
5
What is 80C in Fahrenheit?
5  F  32 
9
What is 80F in Celsius?
Where C is the temperature in Celsius.
Fahrenheit to Celsius
C
 $12229.60
The rate is written as a decimal.
d=rt where d is distance
r is rate and t is time.
Temperature
Celsius to Fahrenheit
(4)(18)
Where F is the temperature in
Fahrenheit.
F
C
9(80)
 32  176 F
5
5 80  32 
 26.67C
9
(Rounded)
3
Statistics
Mean or
average
A
x1  x2  ...  xn
n
Where n is the
A
number of numbers to be averaged
and x1, x2, x3 and so on are the
numbers to be averaged.
Standard
Deviation
 x  x   x
2
s
1
2
 
2

458  500  482  440  500
5
A=476
The standard deviation of Janet’s scores is

2
Janet got a 458, 500, 482, 440, and 500 on her
GED tests. What was her average score?
 x  x3  x  ...  xn  x

 458  476    500  476    482  476    440  476    500  476 
2
2
s
2
2
are the numbers to be averaged.
the average of the list.
x
2
5 1
n 1
Where n is the number of numbers to
be averaged and x1, x2, x3 and so on
2
s=26.5
is
Algebra These are discussed in later chapters, but the formulas can be followed.
Slope
y  y1
where m is slope,
For (3,4) and (-4, 7) the slope is
m 2
x2  x1
m
47
3

3  (4) 7
(x1 , y1) and (x2 , y2) are two
points
Distance
d
x
1
 x2    y1  y2 
2
2
where d is the distance between
two points.
Quadratic
The solution for an equation
0=ax2 + bx + c the solution is
x
b  b 2  4ac
2a
Find the distance between (-2, -3) and (8,3).
d
 2  8    3  3 
2
2

 10    6 
2
 100  36  136  11.66
What are the solutions for:
0=2x2-9x+10 a =2, b=-9 and c=10
(9)  (9) 2  4(2)(10) 9  81  4(2)(10)
x

2(2)
4
9  81  80 9  1 9  1


4
4
4
9 1
9 1
10 5
8

and
for
 and  2
4
4
4 2
4

4
2
Practice: * indicates very challenging problems.
a)
Find the area and perimeter (fringe edge)
of a carpet that measures 14 feet by 12
feet.
Find the area and
perimeter of the
trapezoid.
20 in.
7 in.
10 in.
10 in.
8 in.
b)
Find the volume and surface area of a
can that is 5 inches tall and has a radius
of 3 inches.
26
Find the volume
and surface
area of the
square pyramid
.
m
36 m.
30 m.
30 m.
c)
Change the temperature 21C to
Fahrenheit.
d)
What is the simple interest on $4500 for
2 years at 15%?
e)* What is the accumulated amount for a
$5000 loan compounded monthly for 3
years at 9%?
Change the temperature 72F to Celsius.
What is the simple interest on $4500 for 6
months at 15%?
Note:6 months is ½ a year so t in the
formula is ½.
* What is the accumulated amount for a
$1000 loan compounded semiannually for 20
years at 15%?
Semiannually is 2 times a year so m is 2.
f)
Find the mean of the following list of
numbers.
4,5,3,6,4,8,5,6,5,4
g)
Find the area and circumference of a
circle with a radius of 12 meters.
* Find the standard deviation of the list.
Find the area and
perimeter of the
triangle.
15 in.
21 in.
12 in.
27 in.
h)
Find the retail price of a freezer with a
wholesale price of $350 that is marked up
75%.
Find the retail price of a table with a
wholesale price of $50 that is marked up
175%.
r is 1.75 in the formula.
5
i)
Find the sale price of the freezer in
problem h) The store advertises a
clearance sale of 40% off.
j)
* Find the distance Joe flies at 300 miles
per hour for 2 hours.
* Find the distance a bug crawls at 10 feet
per minute for 7 minutes. He is then
squashed.
k)
Find the surface area and volume of a
cone with a height of 20 cm and a radius
of 12cm. The Slant height is 25 cm.
Find the volume and surface area of a
sphere with a radius of 14cm.
l)
* What is the accumulated amount for a
$800 loan compounded weekly for 5
years at 8%?
* What is the accumulated amount for a
$100 loan compounded semi annually for 15
years at 12%
m)
The wholesale cost of a watch was $85.
What is the retail price with a mark up of
135% (r is 1.35)
If the same table then goes on clearance
and the store offers a 30% discount. (Start
from the answer from the previous problem.)
What is the new cost?
n)
The following are very challenging for the
place we are in this text, but give them a
try.
* Find the slope and distance between
the two points (3,4) and (10, 12)
o)
6
* Find the solution of 0= x2-x-20
a=1, b= -1, and c =-20
Find the sale price of the table in problem h)
The store advertises a clearance sale of 40%
off.
* Find the slope and distance between the
two points (5, -8) and (8, -3)
* Find the solution of 0= x2 –9
a=1, b=0, and c=-9