M 22-24

Highway Engineering

Field Formulas
Metric (SI) or US Units

Unless otherwise stated the formulas shown 

in this manual can be used with any units. 

The user is cautioned not to mix units within 

a formula. Convert all variables to one unit 

system prior to using these formulas.

Significant Digits

Final answers from computations should be 

rounded off to the number of decimal places 

justified by the data. The answer can be no 

more accurate than the least accurate

number in the data. Of course, rounding

should be done on final calculations only. It 

should not be done on interim results.

Persons with disabilities may request this 

information be prepared in alternate forms 

by calling collect (360) 664-9009. Deaf and 

hearing impaired people call 1-800-833-6388 

(TTY Relay Service).

1998

Engineering Publications 

Transportation Building

Olympia, WA 98504

360-705-7430
CONTENTS

Nomenclature for Circular Curves ..................... 2

Circular Curve Equations .................................. 4

Simple Circular Curve ....................................... 5

Degrees of Curvature to Various Radii ............... 6

Nomenclature for Vertical Curves ...................... 7

Vertical Curve Equations ................................... 8

Nomenclature for Nonsymmetrical Curves ......... 10

Nonsymmetrical Vertical Curve Equations ......... 11

Determining Radii of Sharp Curves ................... 12

Dist. from Fin. Shld. to Subgrade Shld. ............. 13

Areas of Plane Figures ..................................... 14

Surfaces and Volumes of Solids ....................... 18

Trigonometric Functions for all Quadrants ........ 23

Trigonometric Functions ................................... 24

Right Triangle .................................................. 25

Oblique Triangle .............................................. 26

Conversion Factors .......................................... 28

Metric Conversion Factors ............................... 30

Land Surveying Conversion Table ................... 31

Steel Tape Temperature Corrections ............... 31

Temperature Conversion ................................. 31

Less Common Conversion Factors .................. 32

Water Constants ............................................. 32

Cement Constants .......................................... 32

Multiplication Factor Table ............................... 33

Recommended Pronunciations ........................ 33

Reinforcing Steel ............................................. 34
2

Nomenclature For

Circular Curves

POT Point On Tangent outside the

effect of any curve

POC Point On a circular Curve

POST Point On a Semi-Tangent (within 

the limits of a curve)

PI Point of Intersection of a back

tangent and forward tangent

PC Point of Curvature - Point of

change from back tangent to

circular curve

PT Point of Tangency - Point of

change from circular curve to

forward tangent

PCC Point of Compound Curvature -

Point common to two curves in the 

same direction with different radii

PRC Point of Reverse Curve - Point

common to two curves in opposite 

directions and with the same or

different radii

L Total Length of any circular curve 

measured along its arc

Lc Length between any two points on 

a circular curve

R Radius of a circular curve

Δ Total intersection (or central) angle 

between back and forward

tangents
3

Nomenclature For 

Circular Curves (Cont.)

DC Deflection angle for full circular

curve measured from tangent at

PC or PT

dc Deflection angle required from

tangent to a circular curve to any 

other point on a circular curve

C Total Chord length, or long chord, 

for a circular curve

C ́ Chord length between any two

points on a circular curve

T Distance along semi-Tangent from 

the point of intersection of the

back and forward tangents to the 

origin of curvature (From the PI to 

the PC or PT)

tx Distance along semi-tangent from 

the PC (or PT) to the perpendicular 

offset to any point on a circular

curve. (Abscissa of any point on a 

circular curve referred to the

beginning of curvature as origin

and semi-tangent as axis)

ty The perpendicular offset, or

ordinate, from the semi-tangent to 

a point on a circular curve

E External distance (radial distance) 

from PI to midpoint on a simple

circular curve
4

Circular Curve Equations

Equations Units

R

L

=

°

×

180

p D

m or ft.

D =

°

×

180

p

L

R

degree

L = ×R

p

180

D m or ft.

T = R tan

D

2

m or ft.

E

R

= - R

cos

D

2

m or ft.

C = 2R or = R DC

2

sin , 2 sin Dm or ft.

MO = R -

æ

è

ç

ö

ø

1 ÷

2

cos

D

m or ft.

DC =

D

2

degree

dc L

L

c =

æ

è

ç

ö

ø

÷

D

2

degree

C' = 2R sin(dc) m or ft.

C = 2R sin(DC) m or ft.

tx = R sin(2dc) m or ft.

ty = R[1 - cos(2dc)] m or ft.
5

Simple Circular Curve

Constant for π = 3.14159265
6

Degree of Curvature for 

Various Lengths of Radii

Exact for Arc Definition

D

R R

=

æ

è

ç

ö

ø

÷

=

100 180

p 18000

p

Where D is Degree of Curvature

__________________________________________

____

Length of Radii for Various 

Degrees of Curvature

R

D D

=

æ

è

ç

ö

ø

÷

=

100 180

p 18000

p

Where R is Radius Length
7

Nomenclature For 

Vertical Curves

G1 & G2 Tangent Grade in percent

A The absolute of the Algebraic

difference in grades in percent

BVC Beginning of Vertical Curve

EVC End of Vertical Curve

VPI Vertical Point of Intersection

L Length of vertical curve

D Horizontal distance to any point on 

the curve from BVC or EVC 

E Vertical distance from VPI to curve

e Vertical distance from any point on 

the curve to the tangent grade

K Distance required to achieve a 1

percent change in grade

L1 Length of a vertical curve which will 

pass through a given point

D0 Distance from the BVC to the

lowest or highest point on curve

X Horizontal distance from P' to VPI

H A point on tangent grade G1 to

vertical position of point P'

P and P' Points on tangent grades
8

Symmetrical Vertical

Curve Equations

A = (G2 ) - (G1 )

E

AL

=

800

E

1

2

Elev.BVC Elev.EVC

2

= Elev.VPI

+

-

æ

è

ç

ö

ø

÷

e

ED

L

=

4

2

2

Notes: All equations use units of length (not 

stations or increments)

The variable A is expressed as an absolute 

in percent (%)

Example: If G1 = +4% and G2 = -2% 

Then A = 6
9

Symmetrical Vertical Curve 

Equations (cont.)

e

AD

L

=

2

200

L

AX e AXe e

A

1

2

2 200 20 100

=

( + + + )

D G

L

A

0 = 1

( )

X

ElevH ElevP

A

=

100 - '

K

L

A

=
10

Nomenclature For 

Nonsymmetrical Vertical 

Curves

G1 & G2 Tangent Grades in percent

A The absolute of the Algebraic

difference in grades in percent

BVC Beginning of Vertical Curve

EVC End of Vertical Curve

VPI Vertical Point of Intersection

l1 Length of first section of vertical 

curve

l2 Length of second section of

vertical curve

L Length of vertical curve

D1 Horizontal distance to any point on 

the curve from BVC towards the

VPI

D2 Horizontal distance to any point on 

the curve from EVC towards the

VPI

e1 Vertical distance from any point on 

the curve to the tangent grade

between BVC and VPI

e2 Vertical distance from any point on 

the curve to the tangent grade

between EVC and VPI

E Vertical distance from VPI to curve
11

Nonsymmetrical Vertical

Curve Equations

A (G ) (G )

L l l

E

l l

l l

A

e m

D

l

e m

D

l

= -

= +

=

+

=

ì

í

î

ü

ý

þ

=

ì

í

î

ü

ý

þ

2 1

1 2

1 2

1 2

1

1

1

2

2

2

2

2

200( )
12

Determining Radii of 

Sharp Curves by Field 

Measurements

R

BC

BD

BD

= +

2

2 2

BC AC=

2

Note: Points A and C may be any two 

points on the curve

Example:

Measure the chord length from A to C

AC = 18.4 then BC = 9.2

Measure the middle ordinate length B to D

BD = 3.5

R = + =

9 2

7 0

3.5

2

13.8

2

.

.
13

Distance From Finished 

Shld. to Subgrade Shld. 

and Slope Equivalents

Equation: x

B

A

=

100

A = Algebraic difference in % between shld. slope 

and subgrade slope

B = Depth of surfacing at finished shoulder

x = Distance from finished shld. to subgrade shld.

Shoulder

Slope

Equivalent

Rate of Grade

Equivalent

Vertical Angle

1:1.5 66.67% 33° 41'24"

1:1.75 57.14% 29° 44'42"

1:2 50.00% 26° 33'54"

1:2.5 40.00% 21° 48'05"

1:3 33.33% 18° 26'06"

1:4 25.00% 14° 02'10"

1:5 20.00% 11° 18'36"

1:6 16.67% 9° 27'44"

1:8 12.50% 7° 07'30"

1:10 10.00% 5° 42'38"

Subgrade

Slope

Equivalent

Rate of Grade

Equivalent

Vertical Angle

.020 / 1 2.00% 1° 08'45"

.025 / 1 2.50% 1° 25'56"

.030 / 1 3.00% 1° 43'06"

.035 / 1 3.50% 2° 00'16"

.040 / 1 4.00% 2° 17'26"

.050 / 1 5.00% 2° 51'45"
14

Areas of Plane Figures

Nomenclature

A = Area h = Height

R = Radius P = Perimeter

__________________________________________

____

Triangle

A

bh

P a b c

=

= + +

2

__________________________________________

____

Circle

A R

P R

=

=

p

p

2

2

__________________________________________

____

Ellipse

A = pab
15

__________________________________________

____
16

Areas of Plane Figures

Segment

A R

R Sin

= p -

2

0

2

360 2

D D

__________________________________________

____

Sector

A R

P R R

=

= +

π

π

2

0

0

360

2

360

2

Δ

Δ

( )

__________________________________________

____

Fillet

A RT R

When A R

= -

æ

è

ç

ö

ø

÷

= =

D

D

360

90 0 2146

0

2

0 2

p

: , .

__________________________________________

____
17

Areas of Plane Figures

Parallelogram

A bh

A ah

P a b

=

=

= +

'

2( )

__________________________________________

____

Trapezoid

A

a b h

=

( + )

2

__________________________________________

____

Polygon

Divide into triangles

A = Sum of all triangles

__________________________________________

____
18

Areas of Plane Figures

Annulus

(Circular Ring)

A = (D - d )

p

4

2 2

__________________________________________

____

Irregular Figure

A L

a j

= b c d e f g h i

+

+ + + + + + + +

æ

è

ç

ö

ø

÷

2

__________________________________________

____
19

Surfaces\Volumes of Solids

Nomenclature

S Lateral surface area

V Volume

A Area of section perpendicular to sides

B Area of base

P Perimeter of base

PA Perimeter of section perpendicular to its 

sides

R Radius of sphere or circle

L Slant height or lateral length

H Perpendicular Height

C Circumference of circle or sphere

__________________________________________

____

Parallelepiped

S = PH S = PAL

V = BH = AL

__________________________________________

____

Pyramid or Cone

Right or Regular

S = PL 1

2

V = BH 1

3

__________________________________________

____
20

Surfaces\Volumes of Solids

Pyramid or Cone, Right or 

Oblique, Regular or Irregular

V = BH 1

3

__________________________________________

____

Prism: Right or Oblique,

Regular or Irregular

S = PH = PAL V = BH = AL

__________________________________________

____

Cylinder: Right or Oblique,

Circular or Elliptic

S = PH = PAL V = BH = AL

__________________________________________

____
21

Surfaces\Volumes of Solids

Frustum of any Prism or Cylinder

V = BH V = A(L + L )

1

2

2 1

__________________________________________

____

Frustum of Pyramid or Cone

Right and Regular, Parallel Ends

S = L(P + p)

1

2

V = H(B+ b + Bb)

1

3

p = perimeter of top b = area of top

__________________________________________

____

Frustum of any Pyramid or Cone, 

with Parallel Ends

V = H(B+ b + Bb)

1

3

b = area of top

__________________________________________

____
22

Surfaces\Volumes of Solids

Sphere

S = 4 R

2

p V = R

4

3

3

p

__________________________________________

____

Spherical Sector

S = R( H + C)

1

2

p 4 V = R H

2

3

2

p

__________________________________________

____

Spherical Segment

S = 2 RH = ( H + C )

1

4

4

2 2

p p

V = H ( R - H)

1

3

3

2

p

____________________________________

__
23

Surfaces\Volumes of Solids

Spherical Zone

S = 2pRH

V = H( + + H )

1

24

3C1 3C 4

2 2 2

p

__________________________________________

____

Circular Ring

S = 4 Rr 2p V = 2 Rr 2 2p

__________________________________________

____

Prismoidal Formula

V ( )

H

= B+ b + M

6

4

M = Area of section parallel to bases,

Midway between them

b = area of top

__________________________________________

____
24

Signs of Trigonometric

Functions for All

Quadrants

Note:

When using a calculator to compute

trigonometric functions from North

Azimuths, the correct sign will be displayed
25

Trigonometric Functions

Sine

Sin y

r

opposite

hypotenuse

q = =

Cosine

cosq = =

x

r

adjacent

hypotenuse

Tangent

tanq = =

y

x

opposite

adjacent

Cotangent

cotq = =

x

y

adjacent

opposite

Secant

sec q = =

r

x

hypotenuse

adjacent

Cosecant

cscq = =

r

y

hypotenuse

opposite

Reciprocal 

Relations

sin

csc

q =

1 tan

cot

q

q

=

1

cos

sec

q =

1

Rectangular

X = r × cosq

y = r ×sinq

Polar

r = (x + y )

2 2

q = arctan y

x

O

P (X,Y)

q

x (adjacent) x

y

(hypotenuse)

 r 

y (opposite )
26

Right Triangles

A+B+C=1800 K=Area

Pythagorean 

Theorem a b c

2 2 2

+ =

A and B are complementary angles

sin A = cos B tan A = cot B sec A = csc B

cos A = sin B cot A = tan B csc A = sec B

Given To 

Find

Equation

a, c A, B, 

b, K

sinA

a

c

= cos B

a

c

= 

b = c - a

2 2

 K

a

= c - a

2

2 2

a, b A, B, 

c, K

tanA

a

b

= tanB

b

a

= 

c = a + b

2 2

 K

ab

=

2

A, a B, b, 

c, K

B = 90 - A

0

 b = a × cotA

c

a

A

=

sin

 k

a A

=

×

2

2

cot

A, b B, a, 

c, K

B = 90 - A

0

 a = b × tan A 

c

b

A

=

cos

 K

b A

=

×

2

2

tan

A, c B, a, 

b, K

B = 90 - A

0

 a = c ×sinA 

b = c× cos A K

c A

=

×

2 2

4

sin

A

S

C

B

c

a

b
27

Oblique Triangles

Law of Sines a

A

b

B

c

sin sin sin C

= =

Law of Cosines

a b c bc A

b a c ac B

c a b ab C

2 2 2

2 2 2

2 2 2

2

2

2

= + - ×

= + - ×

= + - ×

cos

cos

cos

Sum of Angles A + B+ C = 1800

K = Area s

a b c

=

+ +

2

Given

To 

Find Equation

a, b, c A

( )( )

sin A s b s c

2 bc

=

- -

( )

cos

A s s a

2 bc

=

-

( )( )

( )

tan

A s b s c

2 s s a

=

- -

-

c

b

B

C

A

a
28

Oblique Triangles

Given To 

Find Equation

a, b, c B

( )( )

sin B s a s c

2 ac

=

- -

( )

cos

B s s b

2 ac

=

-

( )( )

( )

tan

B s a s c

2 s s b

=

- -

-

a, b, c C

( )( )

sin C s a s b

2 ab

=

- -

( )

cos

C s s c

2 ab

=

-

( )( )

( )

tan

C s a s b

2 s s c

=

- -

-

a, b, c K K = s(s - a)(s - b)(s - c)

a, A, B b, c b

a B

A

=

× sin

sin

 

( )

c

a A B

A

=

× sin +

sin

a, A, B K K

ab C a B C

A

=

×

=

× ×

×

sin sin sin

2 2 sin

2

a, b, A B sin sin B

b A

a

=

×

a, b, A c

c

a C

A

b C

B

=

×

=

sin ×

sin

sin

sin

c = (a + b - ab× C)

2 2

2 cos

a, b, A K K

ab C

=

× sin

2

a, b, C A tan

sin

cos

A

a C

b a C

=

×

- ×

a, b, C c

( )

( )

c

a A B

A

c a b ab C

=

× +

= + - ×

sin

sin

cos 2 2 2

a, b, C K K

ab C

=

× sin

2
29

Conversion Factors

Class multiply: by: to get:

Length in 0.0833 ft

in 0.028 yd

ft 12 in

ft 0.33 yd

ft 0.06 rods

yd 36 in

yd 3 ft

yd 0.18 rods

rods 198 in

rods 16.5 ft

rods 5.5 yd

mi 5280 ft

mi 1760 yd

mi 320 rods

Area in20.007 ft2

ft2144 in2

ft20.11 yd2

yd21296 in2

yd29 ft2

yd20.03 rods2

rods2272.25 ft2

rods230.25 yd2

acres 43560 ft2

acres 4840 yd2

acres 160 rods2
30

Conversion Factors

Class multiply: by: to get:

Volume ft31728 in3

ft30.04 yd3

ft37.48 gallons

yd327 ft3

yd3202 gallons

quarts 2 pints

quarts 0.25 gallons

gallons 8 pints

gallons 4 quarts

gallons 0.13 ft3

Force ounces 0.06 pounds

pounds 16 ounces

tons 

(short)

2000 pounds

tons 

(metric)

2205 pounds

Velocity miles/hr 88 ft/min

miles/hr 1.47 ft/sec
31

Metric Conversion Factors

Class multiply: by: to get:

Length in 25.40 mm

in 2.540 cm

in 0.0254 m

ft 0.3048 m

yd 0.9144 m

mi 1.6093 km

Area ft20.0929 m

2

yd20.8361 m

2

mi22.590 km2

Volume in316.387 cm

3

ft30.0283 m

3

yd30.7646 m

3

gal 3.785 L

gal 0.0038 m

3

fl oz 29.574 mL

acre ft 1233.48 m

3

Mass oz 28.35 g

lb 0.4536 kg

kip

(1000 lb)

0.4536 metric ton

(1000 kg)

short ton

2000 lb

907.2 kg

short ton 0.9072 metric ton
32

Land Surveying

Conversion Factors

Class multiply: by: to get:

Area acre 4046.8726 m

2

acre 0.40469 ha

10000 m2

Length ft 12/39.37* m

* Exact, by definition of the U.S. Survey foot

__________________________________________

____

Steel Tape

Temperature Corrections

C = ⋅ (TC − )Lm

−

1166 10 20 6.

or

C = (TF )Lf⋅ −

−

6 10 68 6.45

Where:

C = Correction

TC = Temperature in degrees Celsius

LM = Length in meters

TF = Temperature in degrees Fahrenheit

Lf = Length in feet

__________________________________________

____

Temperature Conversion

Fahrenheit to Celsius ( )

5

9

°F − 32

Celsius to Fahrenheit 9

5

° 32 









C +

__________________________________________

____
33

Less Common

Conversion Factors

Class multiply: by: to get:

Density lb/ft316.0185 kg/m3

lb/yd30.5933 kg/m3

Pressure psi 6894.8 Pa

ksi 6.8948 MPa

lb/ft247.88 Pa

Velocity ft/s 0.3048 m/s

mph 0.4470 m/s

mph 1.6093 km/h

Water Constants

Freezing point of water = 0° C (32° F)

Boiling point of water under pressure of one 

atmosphere = 100° C (212° F)

The mass of one cu. meter of water is 1000 kg

The mass of one liter of water is 1 kg (2.20 

lbs)

1 cu. ft. of water @60° F = 62.37 lbs (28.29 kg)

1 gal of water @60° F = 8.3377 lbs (3.78 kg)

__________________________________________

____

Cement Constants

1 sack of cement (appx.) = 1 ft3 = 0.028 m3

1 sack of cement = 94 lbs. = 42.64 kg

1 gallon water = 8.3453 lbs. @39.2° F 

1 gallon water = 3.7854 kg @4° C

__________________________________________

____
34

Multiplication Factor Table

Multiple Prefix Symbol

1 000 000 000 = 109giga G

1 000 000 = 106mega M

1 000 = 103kilo k

100 = 102*hecto h

10 = 101*deka da

0.1 = 10-1*deci d

0.01 = 10-2*centi c

0.001 = 10-3milli m

0.000 001 = 10-6micro μ

0.000 000 001 = 10-9nano n

* Avoid when possible

__________________________________________

____

Recommended 

Pronunciations

Prefix Pronunciation

giga jig’a (i as in jig, a as in a-bout

mega as in mega-phone

kilo kill’ oh

hecto heck’ toe

deka deck’ a (a as in a-bout

centi as in centi-pede

milli as in mili-tary

micro as in micro-phone

nano nan’ oh
35

Reinforcing Steel

Bar 

Size

Nominal 

Diameter 

Nominal 

Area

Unit Weight

#3 9.5mm

[0.375 in]

71mm2

[0.110 in2]

0.560kg\m

[0.376 lb\ft]

#4 12.7mm

[0.500 in]

127mm2

[0.197 in2]

0.994kg\m

[0.668 lb\ft]

#5 15.9mm

[0.625 in]

199mm2

[0.309 in2]

1.552kg\m

[1.043 lb\ft]

#6 19.1mm

[0.750 in]

287mm2

[0.445 in2]

2.235kg\m

[1.502 lb\ft]

#7 22.2mm

[0.875 in]

387mm2

[0.600 in2]

3.045kg\m

[2.044 lb\ft]

#8 25.4mm

[1.000 in]

507mm2

[0.786 in2]

3.973kg\m

[2.670 lb\ft]

#9 28.7mm

[1.128 in]

647mm2

[1.003 in2]

5.060kg\m

[3.400 lb\ft]

#10 32.3mm

[1.270 in]

819mm2

[1.270 in2]

6.404kg\m

[4.303 lb\ft]

#11 35.8mm

[1.410 in]

1007mm2

[1.561 in2]

7.907kg\m

[5.313 lb\ft]

#14 43.0mm

[1.693 in]

1452mm2

[2.251 in2]

11.384kg\m

[7.650 lb\ft]

#18 57.3mm

[2.257 in]

2579mm2

[3.998 in2]

20.239kg\m

[13.600 lb\ft]
36

Notes
37

Notes