Engineering Formula Sheet
Statistics
Mode
Mean
Place data in ascending order.
Mode = most frequently occurring value
∑ xi
n
μ=
µ = mean value
Σxi = sum of all data values (x1, x2, x3, …)
n = number of data values
Median
Place data in ascending order.
If n is odd, median = central value
If n is even, median = mean of two central values
Standard Deviation
σ=√
If two values occur at the maximum frequency the
data set is bimodal.
If three or more values occur at the maximum
frequency the data set is multi-modal.
∑(xi - μ)2
n
n = number of data values
σ = standard deviation
xi = individual data value ( x1, x2, x3, …)
𝜇 = mean value
n = number of data values
Range
Range = xmax - xmin
xmax = maximum data value
xmin = minimum data value
Probability
Independent Events
P (A and B and C) = PAPBPC
Frequency
fx =
nx
n
Px =
fx
fa
P (A and B and C) = probability of independent
events A and B and C occurring in sequence
PA = probability of event A
Mutually Exclusive Events
fx = relative frequency of outcome x
nx = number of events with outcome x
n = total number of events
Px = probability of outcome x
fa = frequency of all events
Binomial Probability (order doesn’t matter)
Pk =
n!(pk )(qn-k )
k!(n-k)!
P (A or B) = probability of either mutually exclusive
event A or B occurring in a trial
PA = probability of event A
Σxi = sum of all data values (x1, x2, x3, …)
n = number of data values
Conditional Probability
Pk = binomial probability of k successes in n trials
p = probability of a success
q = 1 – p = probability of failure
k = number of successes
n = number of trials
PLTW, Inc.
P (A or B) = PA + PB
𝑃(𝐴|𝐷) =
𝑃(𝐴) ∙ 𝑃(𝐷|𝐴)
𝑃(𝐴) ∙ 𝑃(𝐷|𝐴) + 𝑃(~𝐴) ∙ 𝑃(𝐷|~𝐴)
P (A|D) = probability of event A given event D
P(A) = probability of event A occurring
P(~A) = probability of event A not occurring
P(D|̶~A) = probability of event D given event A did not occur
Engineering Formulas
Page 1
Plane Geometry
Ellipse
Rectangle
2b
Area = π a b
Circle
Perimeter = 2a + 2b
Area = ab
2a
Circumference =2 π r
Area = π r
B
Triangle
Parallelogram
h
Area = bh
b
c2 = a2 + b2
tan θ =
a
a2 = b2 + c2 – 2bc·cos∠A
b2 = a2 + c2 – 2ac·cos∠B
c2 = a2 + b2 – 2ab·cos∠C
C
Regular Polygons
Right Triangle
sin θ =
Area = ½ bh
Area = n
a
c
c
a
a
A
b
s
s(12 f)
2
f
n = number of sides
θ
𝑏
c
h
b
b
cos θ
c
a
h
Trapezoid
h
h
h
b
b
h
Area = ½(a + b)h
Solid Geometry
Cube
Sphere
Volume = s3
Surface Area = 6s2
s
r
4
s
Volume π r3
3
Surface Area = 4 π r2
s
Rectangular Prism
Cylinder
r
h
Volume = wdh
Surface Area = 2(wd + wh + dh)
d
w
h
Volume = π r2 h
Surface Area = 2 π r h+2 π r2
Right Circular Cone
πr2 h
Volume =
3
Surface Area = π r √r2 +h2
h
Irregular Prism
r
Volume = Ah
h
A = area of base
Pyramid
Volume =
𝐴ℎ
3
A = area of base
PLTW, Inc.
h
Constants
g = 9.8 m/s2 = 32.27 ft/s2
G = 6.67 x 10-11 m3/kg·s2
π = 3.14159
Engineering Formulas
Page 2
Conversions
Mass
Area
Force
1 kg
= 2.205 lbm
1 slug = 32.2 lbm
1 ton = 2000 lbm
1 acre = 4047 m2
= 43,560 ft2
= 0.00156 mi2
1N
1 kip
Energy
= 0.225 lbf
= 1,000 lbf
1J
= 0.239 cal
= 9.48 x 10-4 Btu
= 0.7376 ft·lbf
1kW h = 3,6000,000 J
Pressure
Length
1m
1 km
1 in.
1 mi
1 yd
1 atm
Volume
= 3.28 ft
= 0.621 mi
= 2.54 cm
= 5280 ft
= 3 ft
1L
1mL
= 0.264 gal
= 0.0353 ft3
= 33.8 fl oz
= 1 cm3 = 1 cc
1psi
= 1.01325 bar
= 33.9 ft H2O
= 29.92 in. Hg
= 760 mm Hg
= 101,325 Pa
= 14.7 psi
= 2.31 ft of H2O
Defined Units
1J
1N
1 Pa
1V
1W
1W
1 Hz
1F
1H
Time
Temperature
1K
1d
1h
1 min
1 yr
= 1 ºC
= 1.8 ºF
= 1.8 ºR
= 24 h
= 60 min
= 60 s
= 365 d
Power
1W
= 3.412 Btu/h
= 0.00134 hp
= 14.34 cal/min
= 0.7376 ft·lbf/s
= 1 N· m
= 1 kg·m / s2
= 1 N / m2
=1W/A
=1J/s
=1V/A
= 1 s-1
= 1 A·s / V
= 1 V·s / V
SI Prefixes
Numbers Less Than One
Power of 10
Prefix
Abbreviation
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24
decicentimillimicronanopicofemtoattozeptoyocto-
Equations
d
c
m
µ
n
p
f
a
z
y
Numbers Greater Than One
Power of 10
Prefix
Abbreviation
101
102
103
106
109
1012
1015
1018
1021
1024
Temperature
TK = TC + 273
Mass and Weight
M = VDm
W = mg
W = VDw
V = volume
Dm = mass density
m = mass
Dw = weight density
g = acceleration due to gravity
PLTW, Inc.
TR = TF + 460
TF - 32
TC
=
180
100
TK = temperature in Kelvin
TC = temperature in Celsius
TR = temperature. in Rankin
TF = temperature in Fahrenheit
Engineering Formulas
decahectokiloMegaGigaTeraPetaExaZettaYotta-
da
h
k
M
G
T
P
E
Z
Y
Force
F = ma
F = force
m = mass
a = acceleration
Equations of Static Equilibrium
ΣFx = 0
ΣFy = 0
ΣMP = 0
Fx = force in the x-direction
Fy = force in the y-direction
MP = moment about point P
Page 3
Equations (Continued)
Electricity
Fluid Mechanics
Energy: Work
W = F∙d
W = work
F = force
d = distance
F
A
P = IV
V1
V
RT (series) = R1 + R2+ ··· + Rn
P1
T1
E W
=
t
t
τ∙rpm
P=
5252
P = power
E = energy
W = work
t = time
τ = torque
rpm = revolutions per minute
Efficiency
Efficiency (%) =
= T2 (Charles’ Law)
2
P
= T2 (Guy-Lussanc’s Law)
2
P1V1 = P2V2 (Boyle’s Law)
P=
Pout
∙100%
Pin
Pout = useful power output
Pin = total power input
Energy: Potential
U = mgh
U = potential energy
m =mass
g = acceleration due to gravity
h = height
1
K = 2 mv2
K = kinetic energy
m = mass
v = velocity
Energy: Thermal
Q =mc∆T
Q = thermal energy
m = mass
c = specific heat
∆T = change in temperature
RT (parallel) =
1
1 1
1
+ + ∙∙∙ +
R1 R2
Rn
Kirchhoff’s Current Law
Q = Av
IT = I1 + I2 + ··· + In
n
or IT = ∑k=1 Ik
A1v1 = A2v2
Kirchhoff’s Voltage Law
QP
Horsepower =
1714
absolute pressure = gauge pressure
+ atmospheric pressure
P = absolute pressure
F = Force
A = Area
V = volume
T = absolute temperature
Q = flow rate
v = flow velocity
Mechanics
s=
d
v=
𝐝
t
𝑡
VT = V1 + V2 + ··· + Vn
n
or VT = ∑k=1 Vk
V = voltage
VT = total voltage
I = current
IT = total current
R = resistance
RT = total resistance
P = power
Thermodynamics
P = Q′ = AU∆T
(where acceleration = 0)
P=
Q
∆t
(where acceleration = 0)
U=
1
R
a=
vf − vi
t
X=
vi sin(2θ)
-g
v = v0 + at
Energy: Kinetic
V = IR
P=
T1
Power
Ohm’s Law
d = d0 + v0t + ½at2
v2 = v02 + 2a(d – d0)
τ = dFsinθ
s = speed
v = velocity
a = acceleration
X = range
t = time
d = distance
g = acceleration due to gravity
d = distance
θ = angle
τ = torque
F = force
P = kA
A1v1 = A2v2
Pnet = σAe(T2 4 -T1 4 )
P = rate of heat transfer
Q = thermal energy
A = Area of thermal conductivity
U = coefficient of heat conductivity
(U-factor)
∆T = change in temperature
∆t = change in time
R = resistance to heat flow ( R-value)
k = thermal conductivity
v = velocity
Pnet = net power radiated
W
σ = 5.6696 x 10-8 m2 ∙K4
e = emissivity constant
T1, T2 = temperature at time 1, time 2
v = flow velocity
PLTW, Inc.
Engineering Formulas
POE
DE
Page 4
Section Properties
Moment of Inertia
Rectangle Centroid
h
Ixx
x
x
3
bh
=
12
b
Ixx = moment of inertia of a rectangular section
about x-x axis
x̅ =
∑ Ai
and y̅ =
b
2
and y̅=
h
2
Right Triangle Centroid
x̅=
b
3
and y̅=
h
3
Semi-circle Centroid
Complex Shapes Centroid
∑ xi Ai
x̅=
x̅ = r and y̅=
∑ yi Ai
∑ Ai
x̅= x-distance to the centroid
y̅ = y-distance to the centroid
xi = x distance to centroid of shape i
yi = y distance to centroid of shape i
Ai = Area of shape i
4r
3π
x̅= x-distance to the centroid
y̅ = y-distance to the centroid
Structural Analysis
Material Properties
Beam Formulas
Stress (axial)
Reaction
F
σ=
A
Moment
Deflection
σ = stress
F = axial force
A = cross-sectional area
Reaction
Moment
P
RA = R B =
Mmax =
PL
4
2
(at point of load)
3
PL
Δmax = 48EI
(at point of load)
RA = R B =
2
ωL2
Mmax =
Δmax =
ωL
(at center)
8
5ωL4
384EI
Strain (axial)
Deflection
ϵ= δ
L0
Reaction
Moment
Mmax = Pa (between loads)
ϵ = strain
L0 = original length
δ = change in length
Deflection
Pa
Δmax = 24EI
(3L2 -4a2 )
Reaction
RA =
Moment
Mmax =
Deflection
Δmax = Pab(a+2b)√3a(a+2b)
27EI
Modulus of Elasticity
σ
E=
ε
E=
𝜎(F2 -F1 )L0
(𝛿2 − 𝛿1 )A
E = modulus of elasticity
σ = stress
ε = strain
A = cross-sectional area
F = axial force
δ = deformation
RA = R B = P
Pb
L
and RB =
Pab
L
(at x = √
Truss Analysis
FL0
δ = AE
2J = M + R
Pa
L
3,
when a > 𝑏 )
J = number of joints
M =number of members
R = number of reaction forces
POE
PLTW, Inc.
Engineering Formulas
(at center)
(at Point of Load)
a(a+2b)
Deformation: Axial
δ = deformation
F = axial force
L0 = original length
A = cross-sectional area
E = modulus of elasticity
(at center)
AE 4 CEA 4
Page 5
Simple Machines
Inclined Plane
Mechanical Advantage (MA)
DE
IMA=
DR
% Efficiency= (
FR
AMA=
FE
AMA
) 100
IMA
IMA=
L (slope)
H
Wedge
IMA = Ideal Mechanical Advantage
AMA = Actual Mechanical Advantage
DE = Effort Distance
DR = Resistance Distance
FE = Effort Force
FR = Resistance Force
IMA=
L (⊥ to height)
H
Lever
Screw
1st
Class
IMA =
C
Pitch
Pitch =
2nd
Class
1
TPI
C = Circumference
r = radius
Pitch = distance between
threads
TPI = Threads Per Inch
3rd
Class
Compound Machines
MATOTAL = (MA1) (MA2) (MA3) . . .
Wheel and Axle
Gears; Sprockets with Chains; and Pulleys with
Belts Ratios
-Nout -dout -ωin -τout
GR=
=
=
=
Nin
din ωout τin
Effort at Axle
-dout -ωin -τout
=
=
(pulleys)
din ωout τin
Compound Gears
B
D
GRTOTAL = ( ) ( )
A
C
Effort at Wheel
Pulley Systems
IMA = Total number of strands of a single string
supporting the resistance
IMA =
DE (string pulled)
DR (resistance lifted)
GR = Gear Ratio
ωin = Angular Velocity - driver
ωout = Angular Velocity - driven
Nin = Number of Teeth - driver
Nout = Number of Teeth - driven
din = Diameter - driver
dout = Diameter - driven
τin = Torque - driver
τout = Torque - driven
POE
PLTW, Inc.
Engineering Formulas
Page 6
Structural Design
Steel Beam Design: Shear
Va =
Steel Beam Design: Moment
Vn
Ωv
Ma =
Mn
Ωb
Vn = 0.6FyAw
Mn = FyZx
Va = allowable shear strength
Vn = nominal shear strength
Ωv = 1.5 = factor of safety for shear
Fy = yield stress
Aw = area of web
Ma = allowable bending moment
Mn = nominal moment strength
Ωb = 1.67 = factor of safety for
bending moment
Fy = yield stress
Zx = plastic section modulus about
neutral axis
Storm Water Runoff
Storm Water Drainage
Q = CfCiA
Cc =
𝐶1 𝐴1 + 𝐶2 𝐴2 + ∙∙∙
𝐴1 + 𝐴2 + ∙∙∙
Q = peak storm water runoff rate (ft3/s)
Cf = runoff coefficient adjustment
factor
C = runoff coefficient
i = rainfall intensity (in./h)
A = drainage area (acres)
Runoff Coefficient
Adjustment Factor
Return
Period
Cf
1, 2, 5, 10 1.0
25
1.1
50
1.2
100
1.25
Water Supply
Hazen-Williams Formula
hf =
10.44LQ
C
1.85
1.85 4.8655
d
hf = head loss due to friction (ft of H2O)
L = length of pipe (ft)
Q = water flow rate (gpm)
C = Hazen-Williams constant
d = diameter of pipe (in.)
Dynamic Head
Rational Method Runoff Coefficients
Categorized by Surface
Forested
0.059—0.2
Asphalt
0.7—0.95
Brick
0.7—0.85
Concrete
0.8—0.95
Shingle roof
0.75—0.95
Lawns, well drained (sandy soil)
Up to 2% slope
0.05—0.1
2% to 7% slope
0.10—0.15
Over 7% slope
0.15—0.2
Lawns, poor drainage (clay soil)
Up to 2% slope
0.13—0.17
2% to 7% slope
0.18—0.22
Over 7% slope
0.25—0.35
Driveways,
0.75—0.85
walkways
Categorized by Use
Farmland
0.05—0.3
Pasture
0.05—0.3
Unimproved
0.1—0.3
Parks
0.1—0.25
Cemeteries
0.1—0.25
Railroad yard
0.2—0.40
Playgrounds
0.2—0.35
(except asphalt
or Districts
Business
concrete)
Neighborhood
0.5—0.7
City (downtown)
0.7—0.95
Residential
Single-family
0.3—0.5
Multi-plexes,
0.4—0.6
detached
Multi-plexes,
0.6—0.75
attached
Suburban
0.25—0.4
Apartments,
0.5—0.7
condominiumsIndustrial
Light
0.5—0.8
Heavy
0.6—0.9
Spread Footing Design
qnet = qallowable - pfooting
pfooting = tfooting ∙150
q=
lb
2
ft
P
A
qnet = net allowable soil
bearing pressure
qallowable = total allowable soil
bearing pressure
pfooting = soil bearing pressure
due to footing weight
tfooting = thickness of footing
q = soil bearing pressure
P = column load applied
A = area of footing
dynamic head = static head – head loss
CEA 5
PLTW, Inc.
Engineering Formulas
Page 7
CEA 6
PLTW, Inc.
Engineering Formulas
Page 8
Equivalent Length of (Generic) Fittings
Hazen-Williams Constants
555 Timer Design Equations
T = 0.693 (RA + 2RB)C
f =
1
T
duty-cycle =
(RA +RB )
∙100%
(RA +2RB )
T = period
f = frequency
RA =
RB =
C=
Boolean Algebra
Boolean Theorems
Commutative Law
Consensus Theorems
X• 0 = 0
X•Y = Y•X
̅Y = X + Y
X+X
X•1 = X
X+Y = Y+X
̅Y
̅= X + Y
̅
X+X
X• X =X
Associative Law
̅ =0
X•X
X(YZ) = (XY)Z
X+0=X
X + (Y + Z) = (X + Y) + Z
X+1=1
̅̅̅+ Y
X̅ + XY =̅X
̅ = X̅ + Y
̅
X̅ + XY
DeMorgan’s Theorems
X+X=X
Distributive Law
̅̅̅̅̅
XY = X̅ + ̅
Y
X+̅
X=1
X(Y+Z) = XY + XZ
̅̅̅̅̅̅̅
̅
X+Y = ̅
X•Y
̿=X
X
(X+Y)(W+Z) = XW+XZ+YW+YZ
Speeds and Feeds
N=
CS(12in.
)
ft
πd
fm = ft·nt·N
Plunge Rate = ½ ·fm
N = spindle speed (rpm)
CS = cutting speed (in./min)
d = diameter (in.)
fm = feed rate (in. / min)
ft = feed (in. / tooth)
nt = number of teeth
DE 5 CIM 4
PLTW, Inc.
Engineering Formulas
Page 9
Aerospace Equations
Forces of Flight
CD =
2D
Aρv2
R e=
ρvl
μ
CL =
2L
Aρv2
M = Fd
CL = coefficient of lift
CD = coefficient of drag
L = lift
D = drag
A = wing area
ρ = density
Re = Reynolds number
v = velocity
l = length of fluid travel
μ = fluid viscosity
F = force
m = mass
g = acceleration due to gravity
M = moment
d = moment arm (distance from
datum perpendicular to F)
Propulsion
Orbital Mechanics
F N = W(v j - vo )
I = Fave ∆t
𝑒 =√1 -
Fnet = Favg - Fg
a = vf ∆t
FN = net thrust
W = air mass flow
vo = flight velocity
vj = jet velocity
I = total impulse
Fave = average thrust force
Δt = change in time (thrust
duration)
Fnet = net force
Favg = average force
Fg = force of gravity
vf = final velocity
a = acceleration
Δt = change in time (thrust
duration)
NOTE: Fave and Favg are
easily confused.
T = 2π
F=
b2
a2
a3⁄2
a3⁄2
= 2π
√μ
√GM
GMm
r2
𝑒 = eccentricity
b = semi-minor axis
a =semi-major axis
T = orbital period
a = semi-major axis
μ = gravitational parameter
F = force of gravity between two
bodies
G = universal gravitation constant
M =mass of central body
m = mass of orbiting object
r = distance between center of two
objects
Bernoulli’s Law
Energy
(Ps +
1
K = 2 mv2
U=
PS = static pressure
v = velocity
ρ = density
− GMm
R
E=U + K= −
ρv2
ρv2
) = (Ps +
)
2 1
2 2
GMm
2R
K = kinetic energy
m =mass
v = velocity
U = gravitational potential energy
G = universal gravitation constant
M =mass of central body
m = mass of orbiting object
R = Distance center main body to
center of orbiting object
E = Total Energy of an orbit
Atmosphere Parameters
T =15.04-0.00649h
p = 101.29 [
ρ=
(T+273.1)
]
288.08
5.256
p
0.2869(T+273.1)
T = temperature
h = height
p = pressure
ρ = density
AE 5
PLTW, Inc.
Engineering Formulas
Page 10