How do I calculate a right triangle's angles when I know all three side lengths?

Question

I am building something, and have no idea how to calculate angles. I have a right triangle with all three lengths known. However, I need to know how to the two angles not known.

Answer 1

Since you said "I have no idea how to calculate angles"--and since you are using this for construction, not for learning math--I'll step you through it I'm not trying to be too detailed--but the following gives you specific steps to follow without assuming that you understand the mathematics involved.

What you'll need is a scientific calculator. This is a calculator that has buttons reading "sin," "cos," and "tan." If you don't have one, check with friends, coworkers, etc. If all else failes, write down the calculations (below), then go to Wal-Mart and find one that you can "test" without opening the box. (If you want to go all out, you should be able to buy one for around $15 at Wal-Mart, Office Max, or some place like that.)

Okay, you have a right triangle. You didn't give the lengths, so I'm going to say that your triangle measures 3" on one side, 4" on the second side, and 5" on the third side. The actual dimensions don't really matter for right now--just remember that you will use your shortest measurement everywhere I use "3"--you will use your middle measurement everywhere I use "4"--you will use your longest measurement everywhere I use "5". NOTE: It must be assumed that all of your measurements are in common terms--they are all in inches, all in feet, all in centimeters, etc. It doesn't matter what unit they are measured in, as long as all three are measured in the same units.

The first thing I want you to do on the scientific calculator is find a button--usually in the upper lefthand corner--that says "INV" or "2nd" or "Shift." I will refer to this as the "INV" button. Quite often this button is a different color (blue or yellow) from the others.

Once you have found this, I want you to press the following buttons:
"INV" "sin" "1" "="
IF YOU GOT AN ERROR MESSAGE, YOU ARE USING A SCIENTIFIC CALCULATOR THAT EXPECTS THE INFORMATION ENTERED IN A DIFFERENT FORMAT--FIND ANOTHER ONE. (I could explain how to do it on a variety of different calculators, but that would just take too long. Email me if you really need to know.)

The answer that you got should be either 90 or 1.570796327. If you got 90, you will continue below. IF YOU GOT 1.570796327 AS YOUR ANSWER, THE CALCULATOR IS IN RADIAN MODE. FIND ANOTHER ONE--OR EMAIL ME. (Again, I could explain, but there are too many options here.)

Okay, now we are ready to find your angles--we know that we have a scientific calculator that works the way we want, and the calculator is in degree mode.

Enter the following:
"INV" "tan" "(" "3" "÷" "4" ")" "="
You should get 36.86989765, which is the size of the angle between the sides of the triangle which measure "4" and "5". (Of course, you will be using different figures. Inside the parentheses, you divided your smaller measurement by your medium measurement. The angle which shows up in the display will be the angle between your medium side and your long side.)

For the other angle, enter
"INV" "tan" "(" "4" "÷" "3" ")" "="
This will give you 53.13010235. This is the measurement of the angle (in degrees) between the sides measuring "3" and "5".

Again, this does not explain all of the mathematics--and it does not assume that you know the math. These are step-by-step instructions for finding the specific thing that you are seeking to find.

Answer 2

Calculate Angle Of Right Triangle

Answer 3

Right Triangle Angle Calculator

Answer 4

A triangle cannot have 3 ninety degree angles; it would not be a triangle but rather a rectangle. Why? All three angles of a triangle add up to 180 degrees, and 90 * 3 = 270 which is greater than 180. I like dr bob's answer; however if you draw a triangle on a sphere it will no longer be called a triangle; a triangle is a 2-d object, and that would make it 3-d.

Answer 5

If you have a right triangle, then all you need to get the angles are the lengths of its legs (the two shortest sides) and the inverse tangent (arctan) function.

Say, for example, you have a right triangle with sides 3, 4, and 5.
Arctan (3 / 4) = Arctan (0.75) = approx. 36.87º for the smaller angle.
For the larger angle, you can subtract 90º - 36.87 = approx. 53.13º, or
Arctan (4 / 3) = Arctan (1.3333333...) = approx. 53.13º.

As another example, suppose you have a right triangle with sides 11.4, 35.2, and 37.
Arctan (11.4 / 35.2) = Arctan (0.323863636...) = approx. 17.945º, and the other angle is approx. 72.055º.

This method will only work with right triangles, though. For any triangle, use the Law of Cosines.

Answer 6

SOH CAH TOA

In a right triangle the bottom is called the adjacent. The side that is vertical is called opposite and the long side at an angle is called the hypotenuse. OK. So, refer to my favorite Indian name, SOH CAH TOA

Sin = opposite/hypotenuse

Cos = Adjacent/hypotenuse

Tan = opposite/adjacent

Using this you can find any angle on your right triangle. Especially if you already have the length of all 3 sides.

Hope this helps...

Answer 7

sin(x)= opposite/hypotenuse
cos(x) = adjacent/hypotenuse
tan(x) = opposite/adjacent
Of course, you won't need to use all of the above.
Say the side lengths are 3, 4, 5
You already know the angle opposite 5 is 90 degrees.
5 is the hypotenuse.
3 or 4 could be the opposite side depending on the angle.
To find the angle opposite of the side of length 4:
sin(x) = 4/5
x = arcsin(4/5)
x = 53.13
Since the two angles add to 90, then other is found by taking
90 - 53.13 = 36.87
You could also use
cos(x) = 3/5
x = arccos(3/5) = 53.13
or tan(x) = 4/3
x = arctan(4/3) = 53.13
There are more ways to do this.

Answer 8

since is right angle triangle, use the sine, cos or tan rule.
However calculate one angle is enough(since it is right angle triangle)

Check the rule for the trigonmetry in the book

Answer 9

use these equations
cos alfa = (a^2 - b^2 - c^2) / 2.b.c
cos beta = (b^2 - a^2 - c^2) / 2.a.c
cos gamma = (c^2 - a^2 - b^2) / 2.a.b

you will find 3 angles by using these equations

Answer 10

Use tan (angle) for both angles