Quadratic Equation
Progression
Trigonometric Identities
Spherical Trigonometry
Form:
2
AM ∙ HM = (GM)2
Squared Identities:
2
2
Sine Law:
Ax + Bx + C = 0
Arithmetic Progression:
Roots:
s 2 − 4AC
−B ± √B
x=
2A
Sum of Roots:
B
x1 + x2 = −
A
x1 ∙ x2 = +
C
A
(x + y)n
rth term:
th
= nCm x n−m y m
where: m=r-1
cos 𝑎 = cos 𝑏 cos 𝑐 + sin 𝑏 sin 𝑐 cos 𝐴
1
πR3 E
V = AB H =
3
540°
1
A = bh
2
1
A = ab sin C
2
Square:
Case 1: Unequal rate
rate =
work
time
a+b+c
2
s=
 Clock Problems
Trapezoid
θ=
11M − 60H
2
Ex-circleIn-circle
+ if M is ahead of H
- if M is behind of H
1 1 1 1
= + +
𝑟 𝑟1 𝑟2 𝑟3
Centers of Triangle
INCENTER
- the center of the inscribed circle (incircle)
of the triangle & the point of intersection of
the angle bisectors of the triangle.
Ellipse
a2 + b2
2
A = πab C = 2π√
1
Area = n ∙ R2 sinβ
2
1
Area = n ∙ ah
2
β=
360°
n
16 - hexadecagon
17 - septadecagon
18 - octadecagon
19 - nonadecagon
20 - icosagon
21 - unicosagon
22 - do-icosagon
30 - tricontagon
31 - untricontagon
40 - tetradecagon
50 - quincontagon
60 - hexacontagon
100 - hectogon
1,000 - chilliagon
10,000 - myriagon
1,000,000 - megagon
∞ - aperio (circle)
3 - triangle
4 - quad/tetragon
5 - pentagon
6 - hexagon/sexagon
7 - septagon/heptagon
8 - octagon
9 - nonagon
10 - decagon
11 - undecagon/
monodecagon
12 - dodecagon/
bidecagon
13 - tridecagon
14 - quadridecagon
15 - quindecagon/
pentadecagon
Inscribed Circle:
Cyclic Quadrilateral: (sum of opposite angles=180°)
AT = rs
A = √(s − a)(s − b)(s − c)(s − d)
Escribed Circle:
Ptolemy’s Theorem is applicable:
AT = R a (s − a)
AT = R b (s − b)
AT = R c (s − c)
ac + bd = d1 d2
diameter =
opposite side
sine of angle
a
b
c
=
=
sin A sin B sin C
s=
a+b+c+d
2
Non-cyclic Quadrilateral:
A = √(s − a)(s − b)(s − c)(s − d) − abcd cos 2
Pappus Theorem
Pappus Theorem 1:
Prism or Cylinder
Pointed Solid
SA = L ∙ 2πR
V = AB H = AX L
LA = PB H = Px L
1
V = AB H
3
v
Pappus Theorem 2:
Special Solids
Truncated Prism or Cylinder:
Sphere:
4
V = πR3
3
LA = 4πR2
Frustum of Cone or Pyramid:
Spheroid:
H
(A + A2 + √A1 A2 )
3 1
AB/PB → Perimeter or Area of base
H → Height & L → slant height
AX/PX → Perimeter or Area of crosssection perpendicular to slant height
Spherical Solids
V = AB Have
LA = PB Have
V=
H
V = (A1 + 4AM + A2 )
6
1
2
Spherical Lune:
Spherical Wedge:
Alune 4πR2
=
θrad
2π
3
Vwedge 3 πR
=
θrad
2π
4
2
Vwedge = θR3
Spherical Sector:
1
V = Azone R
3
2
V = πR2 h
3
Spherical Segment:
For one base:
about major axis
4
V = πaab
3
a2 + a2 + b2
]
LA = 4π [
3
LA = PB L
Azone = 2πRh
V = πabb
3
a2 + b2 + b2
]
LA = 4π [
3
Oblate Spheroid:
LA = πrL
Spherical Zone:
4
Prolate Spheroid:
Prismatoid:
Reg. Pyramid
3
V = πabc
3
a2 + b2 + c 2
]
LA = 4π [
3
1
V = πh2 (3R − h)
3
For two bases:
1
about minor axis
V = πh(3a2 + 3b2 + h2 )
6
ε
2
Right Circ. Cone
Alune = 2θR2
4
EULER LINE
- the line that would pass through the
orthocenter, circumcenter, and centroid of
the triangle.
Area = n ∙ ATRIANGLE
δ = 180° − γ
abc
AT =
4R
NOTE: It is also used to locate centroid of an area.
CENTROID
- the point of intersection of the medians of
the triangle.
Deflection Angle, δ:
Circumscribing Circle:
V = A ∙ 2πR
ORTHOCENTER
- the point of intersection of the altitudes of
the triangle.
(n − 2)180°
n
General Quadrilateral
Triangle-Circle Relationship
d=
CIRCUMCENTER
- the center of the circumscribing circle
(circumcircle) & the point of intersection of
the perpendicular bisectors of the triangle.
A = ah
A = a2 sin θ
1
A = d1 d2
2
A1 n
ma2 + nb 2
=
;w = √
A2 m
m+n
1 knot =
1 nautical mile
per hour
Polygon Names
Rhombus:
1
A = (a + b)h
2
1 statute mile =
5280 feet
Central Angle, β:
Parallelogram:
A = √s(s − a)(s − b)(s − c)
Case 2: Equal rate
→ usually in project management
→ express given to man-days or man-hours
γ=
Rectangle:
A = bh
A = ab sin θ
1
A = d1 d2 sin θ
2
1 nautical mile =
6080 feet
Interior Angle, ɤ:
A = s2
A = bh
P = 4s
P = 2a + 2b
d = √2s d = √b 2 + h2
1 sin B sin C
A = a2
2
sin A
 Work Problems
1 minute of arc =
1 nautical mile
n-sided Polygon
2
 Age Problems
→ underline specific time conditions
=0
= vt
180°
sin 2A = 2 sin A cos A
cos 2A = cos 2 A − sin2 A
cos 2A = 2 cos 2 A − 1
cos 2A = 1 − 2 sin2 A
2 tan A
# of diagonals:
tan 2A =
n
1 − tan2 A
d = (n − 3)
Common Quadrilateral
→s
Spherical Polygon:
πR2 E E = spherical excess
AB =
E = (A+B+C+D…) – (n-2)180°
Spherical Pyramid:
Triangle
→a
cos 𝐴 = − cos 𝐵 cos 𝐶 + sin 𝐵 sin 𝐶 cos 𝑎
Double Angle Identities:
Worded Problems Tips
 Motion Problems
Cosine Law for angles:
sin (A ± B) = sin A cos B ± cos A sin B
cos (A ± B) = cos A cos B ∓ sin A sin B
tan A ± tan B
tan (A ± B) =
1 ∓ tan A tan B
r = a 2 /a1 = a 3 /a2
a n = a1 r n−1
a n = a x r n−x
1 − rn
Sn = a1
1−r
a1
S∞ =
1−r
Form:
Cosine Law for sides:
Sum & Diff of Angles Identities:
Geometric Progression:
Binomial Theorem
r
d = a 2 − a1 = a 3 − a 2
a n = a1 + (n − 1)d
a n = a x + (n − x)d
n
Sn = (a1 + a n )
2
Harmonic Progression:
- reciprocal of arithmetic
progression
Product of Roots:
sin 𝑎 sin 𝑏 sin 𝑎
=
=
sin 𝐴 sin 𝐵 sin 𝐴
sin A + cos A = 1
1 + tan2 A = sec 2 A
1 + cot 2 A = csc 2 A
Archimedean Solids
Analytic Geometry
- the only 13 polyhedra that are
convex, have identical vertices, and
their faces are regular polygons.
E=
Nn
2
V=
s
Nn
v
where:
E → # of edges
V → # of vertices
N → # of faces
n → # of sides of each face
v → # of faces meeting at a vertex
Conic Sections
General Equation:
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
Based on discriminant:
B 2 − 4AC = 0 ∴ parabola
B 2 − 4AC < 0 ∴ ellipse
B 2 − 4AC > 0 ∴ hyperbola
Based on eccentricity, e=f/d:
𝑒 = 0 ∴ circle
𝑒 = 1 ∴ parabola
𝑒 < 1 ∴ ellipse
𝑒 > 1 ∴ hyperbola
Distance from a point to another point:
d = √(y2 − y1 )2 + (x2 − x1 )2
Point-slope form:
Distance from a point to a line:
General Equation:
General Equation:
Ax 2 + Cy 2 + Dx + Ey + F = 0
Ax 2 − Cy 2 + Dx + Ey + F = 0
y − y1
m=
x − x1
x 2 + y 2 + Dx + Ey + F = 0
Standard Equation:
2
(x − h) + (y − k)2 = r 2
Two-point form:
y2 − y1 y − y2
=
x 2 − x1 x − x 2
|C1 − C2 |
√A2 + B 2
tan θ =
1 revolution
= 2π rad
= 360˚
= 400 grads
= 6400 mills
2
H = a√
3
√2
V=a
12
Elements:
Eccentricity, e:
e=
3
=1
LR = 4a
Location of foci, c:
c 2 = a2 − b2
Length of LR:
2b2
LR =
a
Versed cosine:
covers A = 1 − sin A
exsec A = sec A − 1
 Inflation:
 Rate of return:
annual net profit
RR =
capital
𝑖f = 𝑖 + f + 𝑖f
 Break-even analysis:
Annual net profit
= savings – expenses
– depreciation (sinking fund)
1
RR
𝑦"
(+) minima
(-) maxima
Integral Calculus-The Cardioid
r = a(1 − cos θ)
r = a(1 + cos θ)
CALTECH:
Mode 3 2
x
y
(time)
(BV)
0
FC
n
SV
(1 + i)n − 1 −1
d = (FC − SV) [
]
𝑖
m
(1 + i) − 1
Dm = d [
]
𝑖
n−m+1
dm = (FC − SV) [
]
∑ years
∑nn−m+1 x
Dm = (FC − SV) [
]
∑n1 x
 Declining Balance (Matheson):
BVm = FC(1 − k)m
SV = FC(1 − k)n k → obtained
Dm = FC − BVm
where:
F → future worth
P → principal or present worth
A → periodic payment
i → interest rate per payment
n → no. of interest periods
n’ → no. of payments
 Perpetuity:
P=
where:
FC → first cost
SV → salvage cost
d → depreciation
per year
n → economic life
m → any year before n
BVm → book value
after m years
Dm → total depreciation
CALTECH:
Mode 3 3
x
y
(time)
(BV)
0
FC
n
SV
n+1
SV
k = 2/n k → obtained
Dm = FC − BVm
 Service Output Method:
A
= F(1 + 𝑖)−n
𝑖
 Capitalized Cost:
C = FC +
OM
RC − SV
+
𝑖
(1 + 𝑖)n − 1
AC = C ∙ 𝑖
AC = FC ∙ 𝑖 + OM +
where:
C → capitalized cost
FC → first cost
OM → annual operation
or maintenance cost
RC → replacement cost
SV → salvage cost
AC → annual cost
(RC − SV)𝑖
(1 + i)n − 1
 Single-payment-compound-amount factor:
n
(F/P, 𝑖, n) = (1 + 𝑖)
 Single-payment-present-worth factor:
−n
(P/F, 𝑖, n) = (1 + 𝑖)
 Equal-payment-series-compound-amount factor:
CALTECH:
Mode 3 6
x
y
(time)
(BV)
0
FC
n
SV
 Double Declining Balance:
BVm = FC(1 − k)m
FC − SV
Qn
D = dQ m
(1 + 𝑖)n − 1
]
𝑖
′
 Sinking Fund:
d=
 Annuity:
(1 + 𝑖)n − 1
P = A[
]
𝑖(1 + 𝑖)n
 Sum-of-the-Years-Digit (SYD):
𝑑2 𝑦
= y" = 0
𝑑𝑥 2
F = Pe
ER = er − 1
′
BVm = FC − Dm
ρ=
where:
F → future worth
P → principal or present worth
i → interest rate per interest period
r → nominal interest rate
n → no. of interest periods
m → no. of interest period per year
t → no. of years
ER → effective rate
 Continuous Compounding Interest:
rt
F = A[
Depreciation
Radius of curvature:
3
[1 + (y′)2 ]2
where:
m is (+) for upward asymptote;
m is (-) for downward
m = b/a if the transverse axis is horizontal;
m = a/b if the transverse axis is vertical
y − k = ±m(x − h)
c
e=
a
F = P(1 + 𝑖)
r mt
F = P (1 + )
m
I
r m
ER = = (1 − ) − 1
P
m
1 − cos A
2
FC − SV
d=
n
Dm = d(m)
Eccentricity, e:
Eq’n of asymptote:
 Compound Interest:
n
Half versed sine:
 Straight-Line:
c 2 = a2 + b2
I = P𝑖n
F = P(1 + 𝑖n)
vers A = 1 − cos A
Exsecant:
Same as ellipse:
Length of LR,
Loc. of directrix, d
Eccentricity, e
a
d=
e
 Simple Interest:
Versed sine:
hav A =
Location of foci, c:
Loc. of directrix, d:
Engineering Economy
Unit Circle
RP =
Point of inflection:
A = 1.5πa2
P = 8a
r = a(1 − sin θ)
r = a(1 + sin θ)
dd
Length of latus
rectum, LR:
cost = revenue
Maxima & Minima (Critical Points):
𝑑𝑦
= y′ = 0
𝑑𝑥
df
Elements:
Elements:
2
Differential Calculus
3
(y′)2 ]2
(y − k)2 (x − h)2
−
=1
a2
b2
m2 − m1
1 + m1 m2
General Equation:
2
𝑥 2 → 𝑥𝑥1
𝑦 2 → 𝑦𝑦1
𝑥 + 𝑥1
𝑥→
2
𝑦 + 𝑦1
𝑦→
2
𝑥𝑦1 + 𝑦𝑥1
𝑥𝑦 →
2
y"
(x − h)2 (y − k)2
+
=1
b2
a2
- the locus of point that moves such that it is always equidistant from a
fixed point (focus) and a fixed line (directrix).
SA = a √3
Curvature:
(x − h)2 (y − k)2
−
=1
a2
b2
Parabola
In the equation of the conic
equation, replace:
[1 +
d=
Standard Equation:
(x − h)2 (y − k)2
+
=1
a2
b2
Angle between two lines:
x y
+ =1
a b
Line Tangent to Conic Section
To find the equation of a line
tangent to a conic section at a
given point P(x1, y1):
Standard Equation:
√A2 + B 2
Distance of two parallel lines:
Point-slope form:
Tetrahedron
k=
d=
|Ax + By + C|
(x − h) = ±4a(y − k)
(y − k)2 = ±4a(x − h)
General Equation:
- the locus of point that moves such
that the difference of its distances
from two fixed points called the foci
is constant.
y = mx + b
Slope-intercept form:
Standard Equation:
2
- the locus of point that moves such
that its distance from a fixed point
called the center is constant.
Hyperbola
- the locus of point that moves such
that the sum of its distances from
two fixed points called the foci is
constant.
y + Dx + Ey + F = 0
x 2 + Dx + Ey + F = 0
Circle
Ellipse
′
(1 + 𝑖)n − 1
(F/A, 𝑖, n) = [
]
𝑖
 Equal-payment-sinking-fund factor:
′
−1
(1 + 𝑖)n − 1
(A/F, 𝑖, n) = [
]
𝑖
 Equal-payment-series-present-worth factor:
′
where:
FC → first cost
SV → salvage cost
d → depreciation per year
Qn → qty produced during
economic life
Qm → qty produced during
up to m year
Dm → total depreciation
(1 + 𝑖)n − 1
(P/A, 𝑖, n) = [
]
𝑖(1 + 𝑖)n
 Equal-payment-series-capital-recovery factor:
′
(1 + 𝑖)n − 1
(A/P, 𝑖, n) = [
]
𝑖(1 + 𝑖)n
−1
Statistics
Fractiles
Transportation Engineering
Traffic Accident Analysis
Measure of Natural Tendency
 Range
= 𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑑𝑎𝑡𝑢𝑚 − 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑑𝑎𝑡𝑢𝑚
Design of Horizontal Curve
 Mean, x̅, μ → average
→ Mode Stat 1-var
 Coefficient of Range
 Accident rate for 100 million
vehicles per miles of travel in a
segment of a highway:
→ Shift Mode ▼s Stat Frequency? on
→ Input
→ AC Shift 1 var x̅
 Quartiles
 Median, Me → middle no.
when n is even
1
n+1
2
1 n
n
= [( ) + ( + 1)]
2 2
2
Q1 = n
Me th =
Me
th
𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑑𝑎𝑡𝑢𝑚 − 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑑𝑎𝑡𝑢𝑚
=
𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑑𝑎𝑡𝑢𝑚 + 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑑𝑎𝑡𝑢𝑚
4
2
3
Q2 = n
Q3 = n
4
4
when n is odd
Q1 =
 Mode, Mo → most frequent
1
1
1
(n + 1) ; Q1 = (n + 1) ; Q1 = (n + 1)
4
4
4
 Interquartile Range, IQR
Standard Deviation
= 𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 − 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒
= Q3 − Q1
 Population standard deviation
→ Mode Stat 1-var
→ Shift Mode ▼ Stat Frequency? on
→ Input
→ AC Shift 1 var σx
 Sample standard deviation
→ Mode Stat 1-var
𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 − 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒
=
𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 + 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒
Q − Q1
= 3
Q3 + Q1
 Outlier
→ extremely high or low data higher than
or lower than the following limits:
NOTE:
If not specified whether population/sample
in a given problem, look for POPULATION.
Q1 − 1.5IQR > x
Q 3 + 1.5IQR < x
Coefficient of Linear Correlation
or Pearson’s r
 Decile or Percentile
→ AC Shift 1 Reg r
R=
R → minimum radius of curvature
e → superelevation
f → coeff. of side friction or
skid resistance
v → design speed in m/s
g → 9.82 m/s2
 Centrifugal ratio or impact factor
2
Impact factor =
v
gR
P = vR
P → power needed to move vehicle in watts
v → velocity of vehicle in m/s
R → sum of diff. resistances in N
Design of Pavement
 Rigid pavement without dowels
t=√
3W
4f
(at the center)
 Flexible pavement
 Population standard deviation
 Z-score or
standard score
or variate
 standard deviation = σ
 variance = σ2
→ Mode Stat
→ AC Shift 1 Distr
left of z → P(
x−μ
z=
σ
 relative variability = σ/x
Mean/Average Deviation
right of z → R(
bet. z & axis → Q(
→ Input
x → no. of observations
μ → mean value, x
̅
σ → standard deviation
 Mean/average value
b
1
mv =
∫ f(x)dx
b−a a
b
1
RMS = √
∫ f(x)2 dx
b−a a
Walli’s Formula
 Binomial Probability Distribution
x n−x
∫ cosm θ sinn θ dθ =
π
2
P(x) = C(n, x) p q
0
where:
p → success
q → failure
f → fatal
i → injury
p → property damage
∑d
n
=
∑t ∑ 1
( )
U1
 Time mean speed, Ut:
d
∑ U1
Ut = t =
n
n
∑
Ʃd → sum of distance traveled by all vehicles
Ʃt → sum of time traveled by all vehicles
Ʃu1 → sum of all spot speed
1/Ʃu1 → reciprocal of sum of all spot speed
n → no. of vehicles
q → rate of flow in vehicles/hour
k → density in vehicles/km
uS → space mean speed in kph
 Thickness of pavement in terms
of expansion pressure
 Minimum time headway (hrs)
= 1/q
expansion pressure
pavement density
Es
SF = √
Ep
3
q = kUs
 Spacing of vehicles (km)
= 1/k
 Peak hour factor (PHF)
= q/qmax
s
[(m − 1)(m − 3)(m − 5) … (1 or 2)][(n − 1)(n − 3)(n − 5) … (1 or 2)]
∙α
(m + n)(m + n − 2)(m + n − 4) … (1 or 2)
α = π/2 for m and n are both even
α =1 otherwise
 Geometric Probability Distribution
x−1
)
Tip to remember:
Fibonacci Numbers
 Poisson Probability Distribution
x −μ
μ e
x!
an =
Period, Amplitude & Frequency
Period (T) → interval over which the graph of
function repeats
Amplitude (A) → greatest distance of any point
on the graph from a horizontal line which passes
halfway between the maximum & minimum
values of the function
Frequency (ω) → no. of repetitions/cycles per unit
of time or 1/T
Period
2π/B
2π/B
π/B
f∙i
f∙i∙p
f1 → allow bearing pressure of subgrade
r → radius of circular area of contact
between wheel load & pavement
NOTE:
Function
y = A sin (Bx + C)
y = A cos (Bx + C)
y = A tan (Bx + C)
SR =
ES → modulus of elasticity of subgrade
EP→ modulus of elasticity of pavement
Discrete Probability Distributions
P(x) = p(q
 Severity ratio, SR:
 Rate of flow:
 Stiffness factor of pavement
P(x ≥ a) = e−λa
P(x ≤ a) = 1 − e−λa
P(a ≤ x ≤ b) = e−λa − e−λb
 Mean value
A → no. of accidents during period of analysis
ADT → average daily traffic entering all legs
N → time period in years
W
t=√
−r
𝜋f1
t=
Exponential Distribution
A (1,000,000)
ADT ∙ N ∙ 365
Us =
(at the edge)
-1 ≤ r ≤ +1; otherwise erroneous
Variance
 Accident rate per million entering
vehicles in an intersection:
 Spacing mean speed, US:
3W
t=√
f
t → thickness of pavement
W → wheel load
f → allow tensile stress of concrete
Normal Distribution
A (100,000,000)
ADT ∙ N ∙ 365 ∙ L
A → no. of accidents during period of analysis
ADT → average daily traffic
N → time period in years
L → length of segment in miles
R=
R → minimum radius of curvature
v → design speed in m/s
g → 9.82 m/s2
3W
t=√
2f
NOTE:
P(x) =
v
g(e + f)
 Rigid pavement with dowels
m
im =
(n)
10 or 100
→ Mode Stat A+Bx
→ Input
R=
Power to move a vehicle
 Coefficient of IQR
 Quartile Deviation (semi-IQR) = IQR/2
→ Shift Mode ▼ Stat Frequency? on
→ Input
→ AC Shift 1 var sx
 Minimum radius of curvature
2
Amplitude
A
A
A
1
√5
n
[(
n
1 + √5
1 − √5
) −(
) ]
2
2
x = r cos θ
y = r sin θ
r = x2 + y2
y
θ = tan−1
x
𝑥2 − 𝑥 − 1 = 0
Mode Eqn 5
𝑥=
1 ± √5
2
measure
too long add
too short subtract
Measurement
Corrections
Due to temperature:
Probable Errors
C = αL(T2 − T1 )
Probable Error (single):
(add/subtract); measured length
(P2 − P1 )L
C=
EA
(subtract only); unsupported length
w 2 L3
24P 2
CD = MD (1 −
∑(x − x̅)
n−1
∑(x − x̅)
Em =
= 0.6745√
n(n − 1)
√n
E
Proportionalities of weight, w:
Due to slope:
(subtract only); measured length
𝑤∝
Normal Tension:
0.204W√AE
1
𝐸2
𝑤∝
1
𝑑
𝑤∝𝑛
Area of Closed Traverse
√PN − P
Error of Closure:
L
H = (g1 + g 2 )
8
L 2
x 2 ( 2)
=
L
y
H 1
Error of Closure
Perimeter
1 acre =
4047 m2
from South
D2
(h − h2 ) − 0.067D1 D2
D1 + D2 1
Stadia Measurement
Leveling
Horizontal:
Elev𝐵 = Elev𝐴 + 𝐵𝑆 − 𝐹𝑆
D = d + (f + c)
𝑓
D = ( )s +C
𝑖
D = Ks + C
Inclined Upward:
Inclined:
Total Error:
Reduction to
Sea Level
CD
MD
=
R
R+h
error/setup = −eBS + eFS
Subtense Bar
Inclined Downward:
error/setup = +eBS − eFS
D = cot
θ
2
eT = error/setup ∙ no. of setups
Double Meridian Distance Method DMD
DMD𝑓𝑖𝑟𝑠𝑡 = Dep𝑓𝑖𝑟𝑠𝑡
DMD𝑛 = DMD𝑛−1 + Dep𝑛−1 + Dep𝑛
DMD𝑙𝑎𝑠𝑡 = −Dep𝑙𝑎𝑠𝑡
2A = Σ(DMD ∙ Lat)
d
[h + hn + 2Σh]
2 1
Double Parallel Distance Method DPD
d
A = [h1 + hn + 2Σh𝑜𝑑𝑑 + 4Σh𝑒𝑣𝑒𝑛 ]
3
Relative Error/Precision:
=
h = h2 +
Simpson’s 1/3 Rule:
= √ΣL2 + ΣD2
Azimuth
hcr = 0.067K 2
Trapezoidal Rule:
A=
Symmetrical:
e
)
TL
Effect of Curvature & Refraction
Area of Irregular Boundaries
Lat = L cos α
Dep = L sin α
Parabolic Curves
e
)
TL
D = Ks cos θ + C
H = D cos θ
V = D sin θ
E=error; d=distance; n=no. of trials
C 2 = S 2 − h2
PN =
CD = MD (1 +
Probable Error (mean):
Due to sag:
C=
E = 0.6745√
too long
too short
(add/subtract); measured length
Due to pull:
lay-out
subtract
add
Note: n must be odd
Simple, Compound & Reverse Curves
DPD𝑓𝑖𝑟𝑠𝑡 = Lat𝑓𝑖𝑟𝑠𝑡
DPD𝑛 = DPD𝑛−1 + Lat 𝑛−1 + Lat 𝑛
DPD𝑙𝑎𝑠𝑡 = −Lat 𝑙𝑎𝑠𝑡
2A = Σ(DMD ∙ Dep)
Spiral Curve
Unsymmetrical:
H=
L1 L2
(g + g 2 )
2(L1 +L2 ) 1
g 3 (L1 +L2 ) = g1 L1 + g 2 L2
Note: Consider signs.
Earthworks
𝑑𝐿 0 𝑑𝑅
±𝑓𝐿 ±𝑓 ±𝑓𝑅
A=
f
w
(d + dR ) + (fL + fR )
2 L
4
T = R tan
I
m = R [1 − cos ]
L = 2R sin
L3
6RLs
L
(c − c2 )(d1 − d2 )
12 1
VP = Ve − Cp
L5
I
Y=L−
2
π
Lc = RI ∙
180°
20 2πR
=
D
360°
1145.916
R=
D
Prismoidal Correction:
40R2 Ls
2
Ls
I
+ (R + p) tan
2
2
I
Es = (R + p) sec − R
2
Ts =
Ls =
Volume (Truncated):
0.036k 3
R
0.0079k 2
R
D
L
=
DC Ls
Σh
= A( )
n
e=
A
(Σh1 + 2Σh2 + 3Σh3 + 4Σh4 )
n
Stopping Sight Distance
Parabolic Summit Curve
v2
S = vt +
2g(f ± G)
a = g(f ± G) (deceleration)
v
(breaking time)
tb =
g(f ± G)
f
Eff =
(100)
fave
L>S
v → speed in m/s
t → perception-reaction time
f → coefficient of friction
G → grade/slope of road
x=
2
L
VP = (A1 + 4Am + A2 )
6
VT =
θ
Ls 2
; p=
3
24R
2
Volume (Prismoidal):
VT = ABase ∙ Have
i=
I
E = R [sec − 1]
L
Ve = (A1 + A2 )
2
CP =
L2 180°
∙
2RLs π
2
I
Volume (End Area):
θ=
L=
A(S)2
200(√h1 + √h2 )
2
L<S
200(√h1 + √h2 )
L = 2(S) −
A
L → length of summit curve
S → sight distance
h1 → height of driver’s eye
h1 = 1.143 m or 3.75 ft
h2 → height of object
h2 = 0.15 m or 0.50 ft
2
LT → long tangent
ST → short tangent
R → radius of simple curve
L → length of spiral from TS to any point
along the spiral
Ls → length of spiral
I → angle of intersection
I c → angle of intersection of the simple
curve
p → length of throw or the distance from
tangent that the circular curve has been
offset
x → offset distance (right angle
distance) from tangent to any point on
the spiral
xc → offset distance (right angle
distance) from tangent to SC
Ec → external distance of the simple
curve
θ → spiral angle from tangent to any
point on the spiral
θS → spiral angle from tangent to SC
i → deflection angle from TS to any point
on the spiral
is → deflection angle from TS to SC
y → distance from TS along the tangent
to any point on the spiral
Parabolic Sag Curve
Underpass Sight Distance
Horizontal Curve
L>S
L>S
L>S
A(S)2
L=
122 + 3.5S
A(S)2
L=
800H
L<S
L<S
122 + 3.5S
L = 2(S) −
A
800H
L = 2(S) −
A
R=
A → algebraic difference
of grades, in percent
L → length of sag curve
S → sight distance
A → algebraic difference of
grades, in percent
L → length of sag curve
L<S
L=
A(K)2
395
H= C−
h1 + h2
2
For passengers comfort,
where K is speed in KPH
R=
S2
8M
L(2S − L)
8M
L → length of horizontal
curve
S → sight distance
R → radius of the curve
M → clearance from the
centerline of the road