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Calculating Polygons
Polygon calculations come up frequently in woodworking.
Finding the angles and dimensions of used in
building multi-sided frames, barrels and drums (to name a
few applications) begins with an understanding to the
geometry of regular (symmetrical) polygons.
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| Figure 1 |
Regular Polygon Shapes:
Calculating the Bevel or Miter Angle of the Parts
In most cases, to build a polygon shape, you'll need to know
the bevel or miter angle necessary to join the sides. To do
that, you'll need to use trigonometric functions in
conjunction with a basic property of the polygon shape.
A regular polygon is an example of a complex shape that
can be thought of as the splicing together of a number of
right triangles. Specifically, a regular polygon with
N sides can be divided into N * 2 "fundamental"
right triangles. The 6-sided polygon in
Figure 1, for example, can be
divided into 12 equally proportioned right
triangles.
The lines that form two sides of each triangle also cut
through the center of the circle that circumscribes the
polygon. Because the lines divide the circle into equal
sections, we know that each triangle will have one acute
angle equal to 360/(N * 2).
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|
Figure 2 |
| For angle a
of triangle AOH in
Figure 2: |
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a
= 360/(6 * 2) |
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=> |
a
= 30 degrees |
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| Angle b
of triangle AOH is the
complement of angle a: |
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b
= 90 - 30 |
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| => |
b
= 60 degrees |
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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. 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
down vertical setting of the blade as a 0 degree setting.
The same would hold true if you are cutting the parts for a
multi-sided frame on a miter saw. Most table saw miter
gauges, on the other hand, treat a square cut as a 90 degree
setting, making angle b the
correct angle setting.
Calculating the Dimensions of a Polygon
So far, we know the how to calculate the acute angles that
make up the 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
of the polygon.
If we begin with dimension D1
of the diameter of the circle that circumscribes the entire
shape and work toward finding the length of the sides, we
can begin the calculations by noticing that the length of
side H of the fundamental right
triangle of every polygon is equal to the overall dimension D1
divided by 2, and also that the length of
side O of the triangle is equal to
length of the sides divided by 2. Knowing
that, we can use the sine function to calculate the length
of the sides.
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| For the
6-sided polygon above, if dimension D1 =
8'' : |
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sin(30) = (S/2)/(8/2) |
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=> |
sin(30) = S/8 |
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=> |
.5 * 8
= S |
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=> |
S =
4'' |
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On the other hand, if we start the process knowing
dimension D2 of the polygon, the
length of each side of the shape can be calculated using the
tangent function:
tan(a) * A = S
Beginning with the length of the sides (S),
we can calculate dimension D1
using the sine function:
sin(a)/S = D1
and similarly, using the tangent function we
can 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. 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 in angle calculations
has shown you that the math used in woodworking
isn't especially complicated. We also hope you noticed
that the calculations we went through are based on only a
few concepts from trigonometry. 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. The more comfortable you are in
applying mathematics to general woodworking situations, the
more free you will be to pursue any project you choose.
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