Length overall (LOA)
is the maximum length of the hull.
Length of waterline (LWL)
is the length of the hull where it enters the water.
Length between perpendiculars (LPP)
is the distance between station 0 and the last station (commonly station 10).
is the side measurement on the hull.
Maximum beam (BMAX) is the widest beam anywhere on the hull.
Beam of the waterline (BWL) is the maximum beam at the design waterline.
Draft (T)
is the distance the hull goes into the water.
The canoe draft (TC) is the draft of the hull without the extended fin keel.
Depth (D)
is the vertical distance from the deepest point of the keel to the sheer line.
DC is without the extended fin keel.
Displacement (D or "Delta")
is the weight of the water displaced by the hull. For a boat freely floating on the surface it is the same as the weight of the ship by Archimedes' Principle.
is the bottom of the hull. The symbol K is used to designate the bottom of the canoe body.
Midship section
is located midway between the fore and aft perpendiculars or the fore and aft ends of the design waterline. The submerged area of the midship section is AM.
Maximum section area may not be the midship section area. The submerged area of the maximum section is denoted AX.
The block coefficient (CB)
is the ratio of the volume occupied by the submerged area of the hull divided by the volume of a block of dimensions L, B, and T.
CB = Volume/LBT
Sheer line
is the intersection between the deck and the hull.
is the vertical distance between the sheer line and the waterline. The sum of the draft (T) and the freeboard is the depth (D).

Lines Drawings

Sheer Plan (Side View)
contains "Bow and Buttock Lines". The centerline plane contains the vertical (or outboard) profile of the ship and is often used as a starting point in ship design.
Body Plan (Front View)
contains "Sections" from which the three dimensional ship may be visualized.

Half-Breadth Plan (Top View)
contains "waterlines." The "Design Waterline" is used to indicate the designed draft.

Stations (or sections)
are lines that represent cuts on the hull that are side-to-side and perpendicular to the centerline.
are horizontal cuts on the hull, designated by their height above the baseline (or Keel).
Bow and Buttock lines
are vertical cuts on the hull parallel to the centerline.

Loci and Centers of the Ship

Center of buoyancy (B)
is the center of gravity of the displaced water. The buoyant force is considered to act through this point. Its longitudinal and vertical positions are denoted by LCB and VCB respectively. The vertical position of the center of buoyancy is an important distance for stability calculations and is indicated as "KB".
Center of Gravity (G)
is generally measured as distance from the keel (KG) and from the midship section (LCG). G must be on the same vertical line as the center of buoyancy.
Metacenter (M)
is the intersection of the vertical through the center of buoyancy of an ship, at a small angle of heel, with the centerline. It is generally measured as a distance from the keel (KM).
Metacentric Height (GM) is the basic measure of stability and is equal to KM-KG.
Metacentric Radius (BM) is I / underwater volume.


Naval Architecture Web Site

Larsson, L. and Eliason, R. (1994). Principles of Yacht Design. Camden Maine: International Marine.


Basic Dimensions

Span (b)
is distance from wing tip to wing tip
Chord (c)
is distance from the leading edge to the trailing edge of a wing.
On some wings, this distance changes as you move from the root to the tip of a wing.
Wing Area (S)
For a rectangular wing, the Area (s) is equal to the span (b) times the chord (c).
For other wing shapes, you have to use the proper formula.
Aspect Ratio (A)
For rectangular wings the aspect ratio is the span divided by the chord.
A = b/c
High aspect ratio wings are long and thin, low aspect ration wings are short and fat.
Wings with higher aspect ratios are more efficient because tip loss is minimized.
For any wing the Aspect Ratio is defined as the span squared divided by area.
A = b2/S

Rectangular Wing

Area (S)
is the base times the height, or in this case
the span (b) times the chord (c)

S = bc

Aspect Ratio (A)
is the span divided by the chord

A = b / c


A wing is 24 inches long (that's the span) and 6 inches wide (that's the chord)

b = 24"

c = 6"

S = bc = 24" x 6" = 144 square inches

A = b / c = 24" / 6" = 4 (no units)

Elliptical Wing

Area (S)
is p times half the span times half the chord at the fuselage.

S = p (b/2)(c/2)

Aspect Ratio (A)
is span squared divided by area.

A = b2/S


An elliptical wing is 24 inches wide and has a chord at the root (next to the fuselage) of 4 inches.

b = 24"

c = 4"

S = p (b/2)(c/2)

For p you can use 3.14

S = 3.14(24"/2)(4"/2) = 75.36 square inches

A = b2/S = 24" x 24" / 75.36 = 576/75.36 = 7.64

Trapezoidal Wing

To find area (S) for a trapezoidal wing
you multiply the span by the average of the chord at the wing tips (ctip) and the chord at the root (croot).

S = b(ctip+croot)/2

Aspect Ratio (A)
is the span squared divided by the area

A = b2/S

For a Trapezoidal Wing, you can also define the Taper Ratio
as ctip divided by croot. For a wing that is not swept back (at the quarter chord) the proper taper ratio is 0.45. A wing without taper (rectangle) should be swept forward 20 degrees. A wing swept back 20 degrees should have a taper ratio of 0.20.


A trapezoidal wing has a span of 18 inches. The chord at the tips is 2 inches, the chord at the root (next to the fuselage) is 4 inches.

S = b(ctip+croot)/2

S = 18"(2"+4")/2 = 18"(6")/2 = 18" x 3" = 54 square inches

A = b2/S = 18" x 18" / 54 sq. inches = 324/54 = 6

Other Shapes

For the area of other shapes, use your geometry to add up the area of shapes that you can recognize.

As with all shapes, the Aspect Ratio (A) is the span squared divided by the area.

English Measure

F=ma and the exigency of the English system of measure.

Force (F)
is a push or pull on an object and is measured in pounds (lbs.)
is measured in feet (ft.)
is measured in seconds (sec.)

In the English system, weight is the standard and mass is the derived value.

Newton's second law is:

F = ma

In the specific case of the weight of a body, the weight (W) is equal to the mass (m) times the acceleration due to gravity (g).

W = mg

lb = m ft/sec2

m = lb-sec2/ft also known as a slug

Specific weight and mass density

W = mg

divide both sides by volume

W/V= (m/V)g

w = rg "rho-gee"

where 'w'
is specific weight
(since the symbol w has other uses, we will normally write rg "rho-gee")
units of w are lbs-ft3
units of r ("rho") are slugs/ft3 or lb-sec2/ft-ft3 or lb-sec2/ft4
units of g are ft/sec2

Representative values

g = 32.174 ft/sec2

r("rho") for salt water = 1.9904 lb sec2/ft4

r("rho") air = 0.0023292 lb sec2/ft4

More specific values of r

Metric equivalent

F = ma

W = mg

Weight is measured in Newtons which are kilogram/m-sec2

mass is measured in Kilograms

g = 9.8 m/sec2

so the weight of a 1 kilogram item at the earths surface will be 9.8 Newtons

Energy and Power

Mechanical Systems

In mechanical systems, energy is transferred by doing work.

Work is force times the distance through which it acts.

W = F X D
the units are ft-lbs (a measure of energy)

Thermal Systems

In thermal systems, energy is measured in British Thermal Units (BTU).

1 BTU is the amount of heat required to raise the temperature of 1 pound of water from 59.5 to 60.5 degrees F.

Since Work and BTU are both energy, there must be a conversion factor.

1 BTU = 778.26 ft-lbs.

Numerical Integration

In engineering, many curves are drawn, rather than being the result of some formula. To find the area under the curve, numerical methods must be used. Three methods available are rectangles, trapezoids, or quadratics (Simpson's rule).

If the curve in question is a curve of areas, the methods may be used to find the volume of 3-D shapes.

To find the area under a curve, divide the X-axis into an odd number of equally spaced sections. The width of the section is S.
Determine the Y coordinate of the curve at each X.


Area = S (Y0 + Y1 + ... + Yn-1)

To increase the accuracy of the rectangular method, reduce the sation spacing (S). As S approaches 0, the accuracy increases. When S becomes 0, this method becomes the calculus.


The trapezoidal rule can be used with any number of stations.

Area = (S/2) (Y0 + 2Y1 + ... + 2Yn-1 + Yn)

Simpson's Rule

Simpson's rule is based on drawing a quadratic equation through three points. In order to use Simpson's rule there must be an odd number of sections.

Area = (S/3) (Y0 + 4Y1 + 2Y2 + 4Y3 + ... + 4Yn-1 + Yn)

Coefficients of Lift and Drag

Coefficients of lift and drag are used to scale up the results of wind tunnel tests. These coefficients are based on Bernoulli's Principle

Coefficient of Lift

Recalling Bernoulli's Principle

PV + ½ mV2 + mgZ = Constant

(along a streamline)

Dividing both sides by Volume and

ignoring altitude differences

P + ½rV2 = Constant

and recalling that Force (F) equals pressure (P) times Area (S)

F + ½rV2S = Constant


F/ ( ½rV2S) + 1 = Constant

The Force in question is Lift

the constant of proportionality is CL

Lift /( ½rSV2) = CL

Coefficient of Drag

Similarly for Drag

Drag /(½rSV2) = CD

Control Surfaces

Control Surfaces are used for passive (vertical and horizontal stabilizers) or active control of the six directions of freedom of an aircraft.

The Horizontal Stabilizer and the Vertical Stabilizer are fixed control surfaces

The Rudder controls yaw (right or left)

Ailerons control roll (bank) and are used to turn

Flaps are used to increase lift for take-off and landing

The Throttle is used to climb or descend

  • Controls engine speed (RPM) and hence thrust
  • Controls rate of climb

The Elevator controls airspeed changes pitch (and hence angle of attack) which changes drag

  • On trailing edge of horizontal stabilizer
  • Operate together
  • Stick back is elevator up and hence nose up

Six Directions of freedom

Rotational Motion

Linear Motion


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