Calculating Areas with Curved Outlines                            

By  Will Kuhnle

Calculating the area of a constant chord or straight tapered wing is easy. But how do you find the area of a wing (or tail) that has a curved outline?  One approach is to divide the curved outline into a number of triangles, segments and / or fillets as in Fig. 1 and 2. You calculate the area of each triangle, segment and fillet and sum up the areas to get the total "wing" area. To do this you need a formula for the area of a triangle, segment and fillet.
 
Area of Triangle

The area of a triangle is simply one half the Base (B) times the Height (H). The formula is:
   
At = (1/2) * B * H where At = Area of Triangle.            

Approximate Area of Segment

The exact area of a segment depends on the shape of the curve, and even for a simple curve, like an arc of a circle, the formula is not simple. But fortunately a reasonable approximation is all we need. The following formulas are within 10%, typically within 5%.

where As = Area of Segment

As = (2/3) * B * H     if angle A is greater than 60 degrees ( if A > 60 ).
As = (3/4) * B * H     if angle A is less than 60 degrees (if A < 60).

B, H and A are as shown in Fig. 3.  The lines n-o and m-o that define angle A are tangent to the curve at the end points, n and m.

Approximate  Area of Fillet

The area of the fillet, the area bounded by the curve and the lines n-o and m-o of Fig. 4, is approximated by the following formula:

Af = (2/3) * B * H where Af = Area of Fillet

The line p-q is drawn tangent to the curve and parallel to n-m. Note that the height H is measured perpendicular to the base B, not along line o-n.

Since the formula for the area of a triangle is exact but the formulas for segments and fillets are approximate, it is desirable that most of the wing area be represented by triangles. If the segment-fillet area is off by 10% but the total area of the segments and fillets is less than 20% of the total wing area, then over all error will be less than 2%, that is 10% of 20%.

Example: Calculation of the area of the horizontal stabilizer of Figure 1.
 

The area is 34.9 square inches. The area of the segments is 23% of the total so the error is probably less than 2.3% or 0.8 square inches.  But just to check, I drew a line from b to c to d to e to f, thus dividing the area into 4 triangles and 4 segments. The area came out to 35.2 +/- 0.25 square inches.

Using Fillets

Figure 2 illustrates the use of fillets. The wing is divided into two triangles, age and cea, and two fillets, dcb and def. To get the wing area, the area of the two fillets are subtracted from the sum of the areas of the two triangles.

Summary

The area of complex shapes can be approximated by dividing the area into a number of simple shapes. The use of triangles, fillets and segments has the following advantages:

Minimum  number of simple areas.  Many wing / stabilizer outlines can be adequately represented by dividing it into only four areas, two triangles and two segments.

Easy to remember formulas.

All the formulas have the same form:
 
(area) = (shape factor) * ( base ) * (height)

And the shape factors are common fractions.

1/2 for triangle
2/3 for fillet
2/3 for segment, A larger than 60 deg.
3/4 for segment, A less than 60 deg.

Simple calculations. Only multiplication and division. No trigonometric, square root or other functions required. Calculations can be easily organized as a "spreadsheet".

Try it, you will like it. 

Click on the thumbnails below to view the figures.

Figure 1

Figure 2

Figure 3

Figure 4