BASIC SHOP MATH
Your math skills may have gotten a bit rusty over the years or maybe math wasn't your favorite subject. Now you have taken up woodworking and find that it is difficult to modify patterns or come up with your own designs. If that describes you then perhaps you could use a review of some basic math principles. I will cover some elementary trigonometry in this article, let me know if you find it useful and whether or not you would like to see future articles related to mathematics in the shop.
Angles are an important measure in most construction projects even if it is only implied in square corners. A square of some type (carpenter's, combination, try, machinist's) can be found in most shops. A square enables you to set or check 90 degree or right angles. Angles are often measured in degrees.
There are 360 degrees in a circle, the second hand on a clock sweeps out 360 degrees as it makes one revolution in a minute. The second hand turns 90 degrees or a right angle each 15 seconds or one quarter of a minute.
A triangle is a figure with three sides and three angles. If one of the angles is 90 degrees, it is a right triangle. Other triangles (acute, obtuse) have all angles less than 90 degrees or one angle greater than 90 degrees. They may be drawn as two right triangles. The sum of all three angles inside a triangle is always 180 degrees. That means that for a right triangle, the sum of the two non-right angles is 90 degrees. Say one of the angles is 45 degrees. Then the other non-right angle must also be 45 degrees (90 - 45 = 45). If one angle is 30 degrees, the other non-right angle is 60 degrees (90 - 30 = 60). Those plastic drafting instruments known as triangles most commonly come in 45-45-90 degrees and 30-60-90 degrees.
Another type of angle measuring instrument is a protractor. Protractors are shaped as full circles or half circles with degrees marked along the edges. The graduated arc used for setting blade tilt on table saws and other woodworking equipment is a type of protractor. Generally, the greater the radius of the protractor, the more precision that it can be set. Unfortunately, the radius for setting many tools is too small for setting accurate angles. Even miter saws can be difficult to set to cut mitered joints without gaps on multi sided objects.
You can lay out an angle with as much precision as you like using trigonometric functions. Sines, cosines, and tangents are three simple functions that can be obtained with great precision on inexpensive calculators. The functions are often abbreviated as sin, cos, and tan.
Suppose you want to lay out a 22.5 degree angle for a miter using a carpenters' square. Find the tangent of 22.5 degrees by keying in 22.5 on a scientific calculator then press the TAN key. The result should be about 0.414213. That will be the ratio of y to x for the angle A as shown in the figure.
tan(22.5) = 0.44213 = Y/X
If you lay out a horizontal distance which we will call X, you can solve for Y by multiplying the tangent by X. The distance X should be as large as possible within reason to get the best accuracy in setting the angle. If X is chosen to be ten inches then the vertical distance Y for a 22.5 degree angle will be 10 times 0.44213 or 4.4213 inches. This is where an accurate dial caliper is useful to measure Y as precisely as you can. Lay out Y and connect the lines with R to obtain the desired angle. You could get even better precision by making X and Y larger. If X was 20 inches the Y would be 8.284 inches.