Understanding Angles - Calculating Polygons
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### Calculating Polygons

Polygon calculations come up frequently in woodworking.Â  Finding the angles and dimensions of used in buildingÂ multi-sided frames, barrels and drums (to name a few applications) begins with an understanding to the geometry of regular (symmetrical)Â polygons.

Figure 1

#### Regular Polygon Shapes

##### CalculatingÂ the Bevel or Miter Angle ofÂ the Parts

In most cases, toÂ build a polygon shape,Â you'll need to know the bevel or miter angle necessary to join the sides.Â To do that, you'll need to useÂ trigonometric functionsÂ in conjunction with a basic property of the polygon shape.

A regular polygon is an example of a complex shape that can beÂ thought of asÂ the splicing togetherÂ of a number of right triangles.Â Specifically, aÂ regular polygon with N sidesÂ can be divided intoÂ N * 2Â "fundamental" right triangles.Â  The 6-sided polygon in Figure 1, for example, can be divided into 12 equally proportioned right triangles.

The lines thatÂ formÂ two sides ofÂ each triangle alsoÂ cut through the center of theÂ circle that circumscribes the polygon. Because the linesÂ divideÂ the circleÂ intoÂ equal sections, weÂ knowÂ that each triangle will haveÂ one acute angleÂ equal to 360/(N * 2).

Figure 2

For angle a ofÂ triangle AOH in Figure 2:Â a = 360/(6 * 2)Â =>aÂ  = 30 degreesÂ Angle b of triangle AOH isÂ the complement ofÂ angle a:Â b =Â  90 - 30Â =>b = 60 degrees

The saw setting necessary to cut the bevel or miter where the joints meet will be either angle a or angle b, depending on the calibration system of the saw you are using.Â  Cutting the beveled edge of barrel or drum staves on a table saw will almost always require you to set the saw at angle a, because nearly all table saw bevel angle scales are calibrated to treat a straight up and downÂ vertical setting of the blade as a 0 degree setting.Â  The same would hold trueÂ if you are cutting theÂ parts forÂ a multi-sided frame on a miter saw.Â  Most table saw miter gauges, on the other hand, treat a square cut as a 90 degree setting,Â making angle b the correctÂ angle setting.

##### Calculating the Dimensions of a Polygon

So far, we know the how to calculate the acute angles that make up theÂ fundamental right triangles of a polygon with N sides. Now, using that information, we can find the dimensions of the parts we'd need to cut to build a polygon shape, based on the overall dimensions of the polygon or, if we wanted to, calculate the overall dimensions of the shape based on the dimension of the sides ofÂ the polygon.

If we beginÂ with dimensionÂ D1 of the diameter of the circle that circumscribes the entire shape and work toward finding the length of the sides,Â we can begin the calculationsÂ by noticing that the length of side H of the fundamental right triangle of every polygon is equal to the overall dimensionÂ D1 divided by 2, and also that the length of side O of the triangle is equal to lengthÂ of the sidesÂ divided by 2. Knowing that, we can use the sine function to calculate the length of the sides.

For the 6-sided polygonÂ above, ifÂ dimensionÂ D1Â = 8''Â :Â sin(30) = (S/2)/(8/2)Â =>sin(30) = S/8Â =>.5 * 8 = SÂ =>S = 4''

On the other hand, if we start the process knowing dimensionÂ D2 of theÂ polygon, the length of each side of the shape can be calculated using the tangent function:

tan(a) * A = S

Beginning withÂ the length of the sides (S), we canÂ calculateÂ dimensionÂ D1 using the sine function:

sin(a)/S =Â D1

and similarly, using the tangent function we canÂ calculate dimension D2:

tan(a)/S = D2

The functions and methods used above can be used to find the angles and dimensions of any regular polygon, regardless of its size or number of sides.Â  And with a little "mathematical creativity," they can be applied to any project where shapes can broken down into right triangles.

#### Using Math in Woodworking

If you're one of the many woodworkers who consistently avoid using mathematics to plan and build projects, we hope that this brief tour of the math used inÂ angle calculations hasÂ shown you thatÂ the math used in woodworking isn'tÂ especiallyÂ complicated. We also hope you noticed thatÂ the calculations we went through are based on only a few concepts from trigonometry.Â  Just about every angle calculation problem you will ever encounter in woodworking can be worked through by applying the the functions an theorems discussed here.Â  The more comfortable you are in applyingÂ mathematics to general woodworking situations, the more free you will be toÂ pursue any project you choose.

posted on April 24, 2013 by Rockler