Math Question: Dimensioning Angles

Cabinetmakers chime in on methods for determining the size of an angled plane. February 8, 2005

I need to know the formula for figuring out the length of an angle if the known dimensions are the thickness of the stock and the specific angle. For example, if a board is 2.25 thick and you cut a 28 degree angle on it, what would the length of said angle be?

Forum Responses
(Cabinetmaking Forum)
From contributor M:

From contributor W:
By thickness, did you mean width, or is that measurement missing? This is one of those things that is probably easier to figure out by just cutting a block and measuring it. 4.79 doesn't sound right.

From contributor R:
The set of steps to get this is:

Tan of 28 = 0.532

2.25 x 0.532 = 1.196

If stock thickness is 2.25 and there is a 28 degree bevel cut on it, then the dimension perpendicular to the 2.25 thickness, which is at 90 degrees to the thickness, will be 1.196 long.

2.25 x 2.25 = 5.063

1.196 x 1.196 = 1.430

5.063 + 1.430 = 6.493

SQRT of 6.493 = 2.548

I'm betting there is a shortcut with fewer steps.

From contributor F:
Based on the question, contributor M is correct. Here's the formula:

a=2.25 && thickness
b=28 && degrees
c=length (in this case= 4.79262255)

converting to radians
a / sin((b * pi) / 180) = c

using degrees
a / sin(b) = c

Based on 2.25" square stock, contributor R is correct.

converting to Radians
sqrt((((a * sin((b * pi) / 180)) / sin(((90-b) * pi) / 180))^2) + (a^2)) = c (in this case c=2.54828261)

using degrees
sqrt((a*sin(b)/sin(90-b))^2+a^2)= c

From contributor A:
Right triangle: compute hypotenuse from angle and opposite leg

sine 28 = .46947
c= 2.25/.46947
c = 4.79264


=sum(2.25/cell =sin is in)

So contributor F's using degrees

a / sin(b) = c

is the same if you have a sine function is some calculator or system.

From contributor F:
You can use Google to run the formulas, but you have to use the radian ones.

Some programs assume you are using radians, which can be confusing if you're plugging in degrees. Being able to actually do the math to get the correct answer and check your function is a good thing in my book. When I was in school, we didn't have no fancy smancy calculators - we had to look it up on a chart, and we liked it.

All of those equations can be done by hand if you have charts.

From contributor P:
I believe the questioner was looking for the length of the right triangle leg opposite the 28 degree angle, which is a more useful number when laying out angles on wood than working with the hypotenuse. First, I always recommend that a craftsman have a sketch of what he/she is solving (less confusion).

Then, it's the old "solve the triangle" using basic trigonometry. I use the tangent button on the calculator 99% of the time, rather than the sine and cosine buttons. Tangent is the length of the opposite leg of your 28 degree angle (which is what we are looking for) divided by the length of the adjacent angle (which is 2.25).

The old way of finding the tangent of a 28 degree angle was looking at the tables in the back of a math book. Now it can be done very easily on a calculator with trig buttons. Punch in 28, then hit tangent. You should get .5317 (rounded off). This is the tangent of a 28 degree angle, not the length of any triangle leg. Multiply .5317 by 2.25, and you should get 1.196 (rounded off). That is the length of the triangle leg opposite of the 28 degree angle. To check your answer, calculate backwards. Opposite leg length (1.196) divided by adjacent leg length (2.25). You should get .5317. Make your calculator work backwards (push the inverse button or something similar to it) then hit TAN. You should get 28.

I recommend that folks new to using trig practice this until they understand it well. If you want to find the hypotenuse, find the length of your right triangle legs first, then do the a square + b square = c square. Less confusing than using sine and cosine on the calculator.

From contributor O:
Having taught Ind Tech for a few years and loving math more years, I eventually realized that many of us make basic trig harder than what it is. Think of tan, sin and cos as names of **ratios**. Most of us handle ratios quite well. Like a previous poster, I use tan more than cos and sin. Draw a right triangle. Note one of the angles that isn’t the right angle. The side of this angle that is not the hypotenuse (side opposite the right angle) is called “adjacent” as it is adjacent (beside) the marked angle. The other side of the triangle (not this adjacent angle or the hypotenuse) is called “opposite” since it’s opposite the marked angle. The ratio of the “opposite” side and the “adjacent” side is named Tangent (Tan) and for any given degree angle will always be the same. Our calculators have these ratios memorized so we don’t have to look them up in the back of the book.

A couple of monikers to help remember the ratios are: Some Old Horse Caught Another Horse Taking Oats Away (note first letter capitalized SOH CAH TOA or sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/hypotenuse ) or pronounced like a politically incorrect Indian name, SOHCAHTOA—soak-a-toe-a.

From contributor F:
If you guys are saying there may be another way, I ain't buying it.;-)

From contributor O:
Nothing of the sort. I'm just trying to enlighten someone who may stumble at the math. Many shop students were frustrated with higher math and I found that when using the concept of ratios, many could get it.

Click here to view full size image.

From contributor F:
So what you're saying is:

2.25 / cos((28 * pi) / 180) = 2.55


or for excel:

=sum(2.25/cell =cosine is in)

c$1=2.25 c$2=28


I still ain't buying it. There just can't be that many different ways!

From contributor O:
I think this molehill is looking like a mountain. What do you mean "that many ways"? There are consistently, through all posters, two answers... 4.79 and 2.55, which is based on which way the 28' angle is the face of the board. There are lots of ways to do lots of things. Math included, as you've posted several formulas. Back to the charts in the back of the book days... I remember having to memorize equivalents for different trig functions. I didn't really know why then and know less why today. :-) But I tried to show the two answers through the two drawings and I'm not the only one with those answers. Let me know where I've confused you.

From contributor A:
For the correct answer to this angle length question, I simply went to my $10.00 CAD program, spent 45 seconds drawing the part and used the dimension function. You do need a basic understanding of math, but I solve problems for engineers all day with simple skills and a different approach.

The comments below were added after this Forum discussion was archived as a Knowledge Base article (add your comment).

Comment from contributor K:
If you were to cut the angle at 45 degrees, you would multiply 2.25 by 1.5, for the simple fact that 45 is half of 90. Do the percentage formula (28 over 90 is equal to x over 100. Multiply 28 times 100 and divide by 90). It is simple - take it from an exhibit builder. Your answer should be 2.9475".