# Understanding the Numbers

Basic design formulas for sailboats

In the preliminary design stage many designers use some basic formulas to help them evaluate the boat they are designing. These formulas indicate general trends only and are refined as soon as the design is entered into the computer and a performance prediction is made. However, these numbers are still useful for the average boat buyer or boat owner to use to evaluate boats.

#### The Displacement/Length Ratio

According to Froude's Law, a boat with a long waterline will sail faster than a boat with a short waterline. That is why a-60 footer will go faster than a 20-footer. On the other hand, a boat with heavy displacement is slower than a boat with a light displacement. This is why a 50,000-pound 50-footer is slower than a 10,000-pound 50-footer on certain points of sail. (In general, a heavier boat is faster upwind in heavy air than a lighter boat because waves slow the lighter boat.)

A designer marries the length and the displacement by using the displacement/length ratio, which states that the displacement in tons divided by a fraction of the waterline length cubed equals a certain number. Written out, the formula is:

(Displacement in pounds/2240) / (LWL/100)3

This number is said to give an idea of the ability of the boat. In general, boats with a displacement/length ratio of under 70 can get on a plane in the right conditions. Most production boats have a ratio of 125 to 250. Boats that have a ratio over 250 tend to be cruisers. Long-distance cruisers may have a ratio over 300 and often up as high as 400, although a boat with that high a displacement/length ratio would be very slow in lighter winds. In order for the displacement/length ratio to give a good idea of any trends, a graph should be made by plotting displacement/length ratio against sail area.

#### Froude's Law

Froude's Law also states that a displacement hulled boat will go faster than the wave length created by the hull as it moves through the water. This wave length is equal to 1.34 x LWL. In other words, a boat with a 25-foot waterline length will go no faster than 6.7 knots. Most boats can be overpowered or may sail down the back of a wave and temporarily exceed this calculated maximum, but they cannot maintain that speed unless they have a low displacement/length ratio when they plane.

#### The Sail Area to Displacement Ratio

This formula indicates how much sail area is available to push each pound of displacement through the water. It is non-dimensional in that sail area, which is in square feet, divided by the displacement divided by 64 to get the volume of displacement raised to the 2/3 power. This reduces the volume of displacement to a squared power so we divide sail area in square feet by another squared number. The formula reads:

Sail area (square feet) / (Displacement/64)2/3

It may also be written as:

Sail area / (Disp/64) .666

The sail area to displacement ratio generally works out to be between 14 and 30, with the highest numbers being the fastest boats in lighter winds. Boats with a high number also tend to accelerate faster and need to be reefed earlier. I generally plot this ratio against waterline length to get a graph that appears to be meaningful. If you wish, you could plot it against LOA + LWL/2 to average the overall and the waterline lengths of a boat.

#### Ballast Ratio

As an indicator of stability in modern boats this number is close to useless unless the hull and keel shapes are close to identical. However, it is useful when you are making a cost comparison of several boats. When used in this manner it gives you an idea of the amount of ballast a boat carries relative to the other materials in the boat. The ratio is:

Ballast x 100 / Displacement

#### Wetted Surface to Sail Area Ratio

Another formula that gives fairly good results is the wetted surface area to sail area ratio. This formula tells you how much wetted surface a boat has relative to its sail area. A boat with a lot of wetted surface is likely to be slow in light winds when sailed against a boat with low wetted surface. (In fact, in winds under 8 to 10 knots this is probably the most important ratio of all.) It is simply:

Sail area in square feet / Wetted surface area in square feet

Typically this ratio is around 2 to 6, with the lower number indicating that a boat has a lot of wetted surface relative to its sail area. A boat with a low number will be a slow light-air boat. The only drawback to this formula is that you will have a hard time getting the wetted surface ratio from the designer or builder.

#### Beam to Length Ratio

The beam to length ratio compares a boat's beam against its length. It serves as an indication of whether one boat is beamier than another for its overall length. For the average user this formula gives an idea of the amount of interior volume a boat has relative to another vessel. The beam to length ratio can be made up in several ways. I prefer to use the following formula:

(LWL + LOA/2) / Max beam

Typically this number is around 5 for a 12-meter boat and may be as low as 2.5 or 3 for fatter hull shapes. The higher the number the better the boat's stability should be and the better the boat's windward ability.

#### Capsize Formula

The capsize formula, which was developed in the aftermath of the Fastnet storm in the Irish Sea indicates a boat's tendency to capsize. Boats with a value of under 2 are less likely to capsize than boats with a higher value. The ratio is calculated:

Beam / (Displacement 1/3 /64)

This ratio assumes that beamy boats are harder to capsize and harder to re-right and that heavier boats are also harder to capsize. Remember that this formula is only an indicator not a hard and fast rule.

#### Fuel to Displacement Ratio

How much fuel should a boat carry? If the boat has a generator it will require more fuel than a boat with no genset. If it is a cruising boat it will typically carry a lot more fuel. I developed a formula to compare the fuel capacity of various boats. It is:

(Fuel in gallons x 7.5 (changes gallons to pounds) x 100) / Displacement

I found that boats with a fuel/displacement ratio of under 1 percent tend to be racers and do not require a lot of fuel. Boats with a ratio of 5 to 7 percent tend to be long-distance cruisers. Most production cruisers that do not go far from shore have a ratio in the 2 to 4 percent range.

#### Fresh Water to Displacement Ratio

This ratio has a similar function to the fuel/displacement ratio, in that it indicates how much water a boat should carry. A long-distance cruiser without a watermaker has a ratio of over 5 percent, while a racing boat often has a ratio under 1%. Production cruisers tend to be around 3 to 5 percent. A boat with a watermaker may have a value as low as 2 percent. The formula is:

(Fresh water in gallons x 8 (changes gallons to pounds) x 100) / Displacement

#### Prismatic Coefficient

The prismatic coefficient gives the designer an idea how full or thin the underwater part of the middle of the boat is relative to the ends. For example, a square-ended barge where the ends of the boat are identical to the midsection has a prismatic coefficient of 1. A sailboat has a prismatic of about half of that number because the ends of the boat are tapered. The number is typically about .51 for a boat intended to be fast in light winds and can be up to .59 for a boat intended to be fast in heavier winds. Unfortunately, it is impossible to calculate this value from numbers given on a brochure. You need to ask the designer what it is.

The coefficient is found by taking the largest transverse area of the hull and multiplying that area by the waterline length. This gives you a box shaped like the largest midship section. The volume of this odd-shaped box is divided into the volume of displacement of the hull. The formula reads:

(Volume of displacement) / Largest sectional area x LWL