How-to: Calculate Triangles
Being able to layout and measure right triangles is one of the most crucial aspects of designing and building something. It seems like I use it at least once on almost everything I build.
Most of us had this in highschool or college, but it never hurts to refresh our memory, or teach it to someone who has never learned it.
What you need...
- A calculator with sin, cos, and tan functions
- A ruler
- A square
- A pencil
- A piece of paper or cardboard.
How sin, cos, and tan work...
These help you figure out missing parts of your triangle. As long as you have two values of a triangle(angles or distances) you can figure out all the remaining values. Below is a chart of how these are used. It won't make sense now, but it will be explained later on. You'll have to refer to this chart throughout the tutorial.
Sin = opposite (divided by) hyp
Cos = adjacent (divided by) hyp
Tan = opposite (divided by) adjacent
hyp is the length of the side across from where your 90 angle is at.
opposite is the length of the side across from an angle that you know or are trying to figure out.
adjacent is the length of the side next to an agle that you know or are trying to figure out.
Let's get started...
Let's say you want to make a template to layout a 30 degree angle. First thing you want to do is to take your square and make a 90 degree angle. You'll then want to pick a number, we'll use 6 for this example, and mark your bottom line at 6" away from your right angle. Now you know you want a 30 degree angle, and you know one of your sides is 6", all you have to do is pick the right function and you can figure out all the remaining dimensions.
Since you know your trying to find a 30 degree angle and you also know the legth of the side that is next to it(or adjacent to it), and your wanting to find the length of the side opposite of the 30 degree angle, you are going to use the
tan.
You'll set-up your equation like this...
tan 30 degrees.....x
---------------------= ----
............1..................6
In order to reduce that to a one line equateion you will cross multiply the values...
1(x) = tan 30 degrees(6)
1(x) can be reduced to simply x
x = tan 30 degrees(6)
This is saying that x(the side you are trying to figure out) is equal to the tangent of 30 degrees multiplied by 6. Now you'll want to get out your calculator and type in
30. Next, press the
tan button. This should give you a number .577735... next multiply this number by
6. You'll end up with 3.46. That is approximately 3-7/16.
Now go back to your drawing and measure up 3-7/16". Connect the two lines and you now have a 30 degree angle. Also since all the angles inside a triangle always equal 180, you can figure out the other angle in the upper left. 180 - 90 - 30 = 60 degrees.
Let's move onto another problem...
Let's say you know the length of two sides of a right triangle, but you don't know any of the angles.
Pick which angle you want to figure out. We'll do the upper left one this time. This will be your
x. Since x is the angle you are trying to figure out and you know the adjacent side the hyp side(remember the hyp is the side across from the 90 degree angle), you will use the
cos function. Set-up your equation as follows...
cos x degrees......8
------------------- = ----
..........1..................12
Next you'll cross multiply the values...
cos x degrees(12) = 8(1)
or
cos x degrees(12) = 8
Now you have to isolate the variable so you'll divide both sides by 12.
cos x degrees = 8/12
or
cos x degrees = 2/3
Now we still need to get the x by itself, so instead of using the
cos function on the calculator, we'll use the
inverse cos function. The inverse buttons are generally the same buttons on the calculator, but they have a
-1 behind them. On this calculator they are in the "second key" range.
Type in
2 divide by 3. You'll end up with .66666666... Now hit the
cos-1 button and you'll get
48.19 If your not planning on sending your truck to the moon, 48 degrees will be close enough. You now know that the upper left hand corner is equal to 48 degrees. Also using the formula from before 180 - 90 - 48 = 42 for the lower right hand corner.
I also recomenned checking another corner or dimension with this new number you got. This will insure that you did it properly. We will use the known angle of 42 degrees in the lower right hand corner and check to see if equals the two sides of 8 and 12. For this we know the opposite and the hyp. So we will use the sin button.
sin 42 degrees = 8/12
sin 42 degrees = .66666666
Type in 42, then hit the sin button and you'll get .669. Close enough, remember we rounded that 48.19 earlier.
That basically sums up the sin, cos, and tan functions as well as the sin-1, cos-1, and tan-1 fucntions. Each one is used similiarly and as long as you follow the same basic rules each time, it will help you figure out any angle or dimension of a triangle.
One other thing that may help you out...
If you know the length of two sides of a right triangle, you can find the third by using this simple formula
a squared + b squared = c squared
a and b are the two legs of the triangle
c is always the hyp of the triangle
We'll use that last example where we knew a=8 and c=12. Plug them into the formula...
8 squared + b squared = 12 squared
64 + b squared = 144
Subtract 64 from both sides
b squared = 80
Square root both sides
b = square root of 80
b = 8.94
Let's double check that with the tan function we we're using before. We'll use the 48.19 degree corner in the upper left and we're checking oppposite(x) and we know adjacent(8)
Set-up your equation
tan 48.19 degrees
.......x
-------------------------- = ---
.............1
.........................8
tan 48.19 degrees(8) = x
Plug that into your calculator
x= 8.94
Everyhting checks out.
Conclusion
I hope you have a basic understanding of how all of this works. It's something that gets very easy the more you do it. I've been doing this for quite a while so sometimes I skip basic steps, so if you need something explained more clearly, I'll do my best.