Pythagorean
Theorem
The
Pythagorean
Theorem has been around a long time.
Remember this formula -- A2 + B2 = C2.
It is used to calculate the unknown length of a side of a
right triangle given the other two sides.
This
is very useful, especially in turning a partial arc as illustrated
below.
This
formula is simpler to use and understand then trigonometry.
It is a handy addition to your mathematical bag of tricks,
just remember the
3-4-5
rule.
Let’s
look at this example. This
part has a partial arc connecting the 2.625 diameter to the 4.725
diameter. To program
this we need to know the exact point at which the partial arc
begins. This is the
tangent point of the arc (the exact point where the straight line
ends and the curvy line begins).
Calculating
the beginning point of the partial arc
Step
1.
Calculate the X axis position.
The
print shows us the radius size is 1.75”.
To program this point we need the X position point for the
center of the radius. As
the print shows this is the 2.625 diameter plus both sides of the
radius.
2.625
+ (2 x 1.75) = 6.125 This
is the X diameter position for the start of the partial arc.
Step
2.
Calculate the Z axis position.
We
need to find the Z axis length from the start of the partial arc to
the end of the partial arc. We’ll
use the Pythagorean Theorem here.
Let’s first sketch the triangle.
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Label
the triangle, remember that side C is always the hypotenuse.
Use the Pythagorean formula to find the length of side B.
Side A is determined using the print dimensions.
First
calculate the difference in part diameters.
4.725
– 2.625 = 2.100
Then
calculate the difference per side (halve it).
2.100 ÷ 2 = 1.050
This
is the length of the step between the two diameters.
Subtract this distance from the length of the radius to
determine the length of side A.
1.750
– 1.050 = 0.700 This
is the length of side A.
Now
we know the length of side A and the length of side C (remember this
is the radius length. Side C = 1.75”).
The
Pythagorean formula is A2
+ B2 = C2, transpose the formula to
calculate side B.
C2
- A2 = B2.
Take the square root of B2 to find B.
1.752
– 0.7002
= B2
3.0625
– 0.4900 = B2
2.5725
= B2
B
= 1.6039
Go
back to the print with the sketch of the triangle.
We now know the Z length of the partial arc and can calculate
the program start point of the arc.
8.800
- 1.6039 = 7.1961”
Unlike
trigonometry, the Pythagorean Theorem does not need angle sizes to
calculate the side lengths.
With a basic calculator the square and square root buttons
are all you need.
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