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UnitMath ( Angle Notes )

Last Modified: 12/7/1999

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This page addresses several of the practical considerations when cutting angles with standard wood working tools. In the hip roof and pyramid examples angles are found, but actual construction requires some more information.
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The following figure shows a board with the angle we want to cut

The next figures show how the gauge would have to be set to make this cut. Note the setup on the right is not recommended.

The next figure shows what most saw gauges measure ( the angle to the normal ).

`
Gauge angle = 90 deg - Desired angle `

Desired angle = 90 deg - Gauge angle

With this in mind you may want to add another angle in the output like:

`HipAngleGauge: 90 deg - HipAngle`.

### Measure Distances Rather Than Angles TOP

It is generally not very accurate nor easy to measure angles, but distances are relatively easy, so go with your strength. The following figure shows the various lengths we can measure. Basic trigonometry gives us:
`
sin( angle ) = width / hypotenuse`

cos( angle ) = x / hypotenuse

tan( angle ) = width / x

So given the width of the board and the length of the hypotenuse we can find the cut angle. You will notice that the angle below is not exact, that is because our lengths are not exact. Longer distances will give more precise results.

`
width: 4 inch + 1/2 in pm 1/32 inch;`

hypotenuse: 7 inch + 3/8 inch pm 1/32 inch;

angle: asin( width / hypotenuse ) as deg; ~ ( 37.1 to 38.1 ) deg

If the piece we are cutting is very wide it much easier to draw a cut line and use a portable saw. For this we would calculate x. The following calculation is for a 4 foot wide piece of plywood with a 30 degree angle.

`x: 4 ft / tan ( 30 deg ) as inch & in/16 ~ ( 83 inch + 2.2 in/16 )`

If you hold to those tolerances you can get very accurate angles. For example the above cut is precise to better than a tenth of a degree.

`atan((4 ft pm 1/32 inch) / ( 83 inch + 2.2 in/16 pm 1/32 inch )) as deg ~ 30.0 deg `

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