Associated Topics || Dr. Math Home || Search Dr. Math

### Sides of a 30-60-90 Triangle

Date: 03/13/98 at 11:03:56
From: Susan Totten
Subject: Math Question

If there is a 30:60:90 triangle, and the short side is X, then why
does the hypotenuse equal two X, and the long side equal X times the
square root of 3? Derive this in two ways.

Date: 03/16/98 at 15:44:48
From: Doctor Sorelle
Subject: Re: Math Question

Dear Susan,

I'm going to show you one way of deriving this. Maybe from this you'll
get an idea of how to do it in another way.

If you put two 30-60-90 triangles back to back, you get an equilateral
triangle with a line down the middle perpendicular (at a right angle)
to the base of the triangle.

/|\
/ | \
/  |  \
/   |   \
/____|____\

If you say that each of the sides of the triangle is 2 units (you
could also call this 2X), then the base will also be two units. The
line down the middle of the triangle perpendicular to the base divides
the base into two equal parts, each with a length of 1 unit. So now
that we know that half of the base is 1 unit and any other side is 2
units let's separate the triangles again.

/|
/ |
2  /  |
/   | ? units
/    |
/_____|
1

We now know 2 of the sides' lengths, so we can use the Pythagorean
theorem (a^2+b^2 = c^2) to find the 3rd side.

Think you can do that on your own?

To do the other proof you might consider using the area of a triangle,
Pythagorean's theorem, equilateral triangles, or other things.

Good luck - and if you need more help, please write back!

-Doctor Sorelle,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/

Associated Topics:
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search