By convention, sails are usually measured as either three-sided sails, or four-sided sails. Multi-faceted bat-wings seem to have more sides than that, but they are measured as three-sided sails - as if they had no roach and the leech was a straight line from the peak, at the head, to the clew, near the end of the boom.

This, inner triangle (peak to clew to tack and back to peak) is where the sail gets most of it's power and upwind capability and since a rig's efficiency is generally determined by it's ability to sail to weather, this is the most important part of the sail.

On a bat-wing, where so much of the sail is roach, you could probably justify measuring the actual sections and adding them-up. You will be trying to counterbalance all of that area while reaching so it may be a better choice for determining how much sail to put on the canoe, but for design on paper, we'll follow tradition and use the standard, measurement triangle.

There are two methods for measuring triangular sails. On a sail where the boom is more-or-less perpendicular to the mast, we can multiply the luff measurement (tack to peak, along the mast) by the foot measurement (tack to clew, along the boom) and divide by two.

Some triangular sails, like lateens, leg-o-muttons and jibs don't have those convenient, right angles. We use a slightly different measurement system for them. Since there is no existing, right angle, we add one by drawing a line, perpendicular to the luff (leading edge) from the luff to the clew (back corner). This is called the "Luff Perpendicular" or L.P. Then we multiply the luff measurement by our L.P. and divide the result by two, giving us the sail's area.

The standard, measurement triangle of this bat-wing ignores the roach, as if it was a normal, three-sided sail. With the mast and boom at nearly a right angle, sail area for the triangle is easily calculated - luff x foot, divided by two. Sailmakers label the mainsail luff "dimension P" and foot length, along the boom "dimension E" . Mainsail area is

P x E 2As long as the jog around the joint in the mast isn't too big and the upper luff is vertical, we can usually treat it like one, continuous edge.

E

P

On three-sided sails without convenient, right angles, like lateens and jibs, we draw a line called a "luff perpendicular" from the luff to the clew. Sail area is then calculated as Luff x L.P., divided by two. Any hollow in the leech, as shown here, is ignored for measurement

90

L.P. Luff

Calculating Sail Area for Three-Sided Sails

Area = Luff x L.P. 2

Sail Area on Paper

Scaling Up or Down

What if we have designed a pleasing shape for our sail, calculated it's area, and find that it is larger or smaller than we had planned to put on the canoe, or we also want to have a matching mizzen with the same shape, but smaller? How do we scale the sail up or down?

All we have to do is get-out our handy, pocket calculator and start multiplying all the sail's dimensions - but, there is a catch. Though the edge measurements of the sail increase in a one-dimensional way ( a six-foot-long luff, multiplied by 150% is nine feet long), the sail area is a two dimensional product. Remember those "luff x L.P. = area" calculations?

If our sail area is 30 sq. ft. and we multiply all the edge measurements by 150%, our spars may only need to be 50% longer, but our sail area will more than double, because the sail is growing in two dimensions. The new sail area will be 67.5 sq. ft! If our original intention was to boost sail area from thirty to around forty-five square feet, we seem to have over-shot the mark.

To generate the 67.5 sq. ft. figure, I multiplied the 30 sq. ft. sail area by 150% - twice, once for each dimension. If you distrust calculators, you can also draw it out and measure the shape, but the answer is the same. Obviously, if we want to convert our sail to around 45 sq. ft., we will need to multiply all of our dimensions by a smaller number.

I'm sure that there is a formula for this, and I'm sure that somebody knows it, but I'm also sure that it's not me, so I use the hunt-and-peck method for arriving at the desired area. Back to the calculator.

Thirty square feet, multiplied by, say, 125% - twice, is 46.87 sq. ft. - close, but lets try again. How about using 123% and multiplying twice? This time we end-up with 45.38 sq ft.

Four-sided sails, like lugsails, gaff and spritsails are a little more complex to measure, but use the same, basic methods. By drawing a diagonal line from sail's clew to it's throat, we can subdivide the sail into two triangular sections. Each of these is then measured, as if it were a three-sided sail, and the results are added together. I like the "L.P. method" for doing this type of measurement, but usually draw my perpendiculars out from whichever side of the triangle is the longest.

dividing line

perpendicular for upper triangle

perpendicular for lower triangle

leech

foot

luff

head

upper perp. x leech length 2

= area of upper triangle

lower perp. x dividing line's length 2

= area of lower triangle

Upper Triangle plus Lower Triangle = Total Area

Sail Area Calculations for Four-Sided Sails

76 - Lines on Paper

Finding the Center of Effort

We could start using fractions of percentage points and eventually arrive at 45 sq. ft. but it's not that critical, and I hate multiplying fractions, so we will settle for 45.38 sq. ft. and get back to work. We now know that by increasing our sail's edge dimensions by 123%, we will end-up with a sail that is about the size we need.

If we wanted to shrink our sail, because it was too big, or we wanted a matching mizzen, say, 20 sq. ft., we could hunt-and-peck percentages (multiplying twice, for area again) and eventually find that reducing the edge dimensions to 82% of their original size would yield a 20.17 sq. ft. sail. Close enough.

Though my mathematical techniques are simplistic - bordering on moronic, they get the job done and are nearly as much fun as my physics experiments (banana stomping).

Now that we have calculated our sail area, adjusted it, as necessary, located the best places in the canoe to step the masts and drawn the sails accurately on the plan, we are ready to find the Center of Effort, that theoretical point where the wind's power is focused upon the sailplan.

This is actually pretty simple, involving no noticeable math or physics. We start by reverting to our basic, measurement triangles. For four-sided sails, we will again mentally divide them into the two triangles that we used to measure their area.

On a three-sided sail, like a typical mainsail, or a lateen, we use the length of the luff and the leech. Put a mark at the midpoint of each of these edges. Now draw two lines: one from the clew to the luff's midpoint, and the other from the tack to the leech's mid point.The spot where these lines cross, is the Center of Effort for that sail. Here again, for design purposes, we ignore roach and leech hollow.

Luff

Leech

Foot ClewCorner

Tack Corner

Leech Midpoint

Luff Midpoint

C. E.

Leech

Foot

Head

On our four-sided sail, we start by finding the C.E. of both triangles. One is upside-down, but that doesn't matter. We use the dividing line as one of our long edges and it's midpoint can be used for drawing lines on both upper and lower sections.We now have a C.E. for each triangle.

Luff

step 1

77 - Lines on Paper

78 - Lines on Paper

step 2

connecting the centers

21.45 sq ft.

17.3 sq. ft. step 3

Now draw a line, connecting the two centers. In order to find the C.E. for the sail, as a whole, we will need to average these two sections, using their relative sail areas. The "C.E. Total" will lie somewhere along the line between the centers of the sections, but where?

When we calculated the sails area, we first calculated each, individual triangle. Those are the measurements we will use to locate the final position of the sail's Center of Effort.

When I originally designed this sail, it ended-up being 38.75 sq. ft. total, with 17.3 sq. ft. in the upper triangle and 21.45 sq. ft. in the lower one. Since we already have those figures, we will use them for this exercise.

We want to draw two more lines, at right angles to the line connecting the centers. Each starts at one of the centers and they head-off in opposite directions. It doesn't matter which one goes which direction. The result should look

like step 3. The length of these lines is important. The length of the upper one, is based on the area of the LOWER triangle and the lower one represents the area of the UPPER triangle.

You can use inches, feet, Astronomical Units or pumpkin diameters, whatever will fit on your page, as long as, in this case, the upper line is 21.45 "units" long, and the lower one is 17.3 units long.

Finally, we connect the outer ends of these lines to each other. The new line crosses the line that connected our individual centers, from step 2, and the spot where the new line crosses the old line is the actual Center of Effort for the entire sail. This now becomes the only C.E. that we will use for any, future calculations having to do with this sail's C.E. The finished product is visible in the step 4 drawing.

perpendiculars added

step 4

17.3 units21.45 sq ft.

17.3 sq. ft.

C.E. Total 38.75 sq. ft.

38.75 sq. ft.

The finished drawing of this four-sided sail, with it's C.E. marked, is ready to insert into a sailplan. Detail, such as corner patches, grommets and trim can be added as desired.

For a twin-sailed rig, we place the individual sails in their proper locations on the plan and then use almost the identical technique to find the C.E. for the combined pair.Once we have each sail's area and have marked it's C.E., a line is drawn between the two centers, perpendiculars are drawn, out from each sail's C.E. and their lengths reflect the size of the opposing sail in some convenient unit of measurement.

The tips of the perpendiculars are connected and the spot where that line cross the line connecting the centers is the total Center of Effort for the sailplan.

This same technique can be used to find the C.E. of a big schooner with multiple masts, several jibs and a combination of three and four-sided sails. Each is measured individually, then the centers are found and combined into sub-groups. These are then combined into larger groups until finally, only two remain. Their combined center is the boat's total, combined C.E.....but it's a lot of work.

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MADISON,

Wisconsin

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MADISON,

Wisconsin

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Individual and combined Centers of Effort on a Twin-Lateen Rig

21.45 units

79 - Lines on Paper