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### Angle, Side Length of a Triangle

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Date: 9/4/96 at 17:19:5
From: Robert Inman
Subject: Angle, Side Length of a Triangle

Hi Dr. Math,

What is the relation between the angles of a triangle and the length
of its sides? For example, if I know that a triangle has a 90 degree
angle, I know the length of one leg, and the angle between that leg
and the hypotenuse, how do I determine the length of the other leg?
```

```
Date: 9/4/96 at 18:53:10
From: Doctor Tom
Subject: Re: Angle, Side Length of a Triangle

Hi Robert,

One simple relation is that the bigger the angle, the bigger the side
opposite it.

There's an exact relation called the "law of sines".  Assume that your
triangle has sides of length A, B, and C.  Assume also that the angle
opposite side A is a, opposite B is b, and opposite C is c.  The law
of sines states:

sin a   sin b   sin c
----- = ----- = -----
A       B       C

Unfortunately, it uses trigonometry.  You can look up the values of
the sines of angles in tables, or using some calculators.

The "law of cosines" is also sometimes valuable.  It states that:

C^2 = A^2 + B^2 - 2ABcos c

with the same labels as above.  "cos" is the cosine function, and
I use "^" to indicate an exponent: "C^2" is "C squared".

"Trigonometry" comes from the Greek: "trig" - triangle, and
"metry" - measurement.  That's exactly what you're trying to
do, right?

-Doctor Tom,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons
High School Trigonometry
Middle School Geometry
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

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