Hamilton's Math To Build On - copyright 1993

Ratio of the Sides of a Right Triangle

About Math To Build On || Contents || On to Function Memory Aid || Back to Names of Sides || Glossary

* Ratio of the Sides of a Right Triangle

A ratio is a comparison by division. You divide one factor into another to see how they compare.

In sports, ratios are constantly flashed on the TV screen. With baseball, we are shown a player's batting average, which compares his number of hits to his number of times at bat.

In basketball, a ratio of shots made to shots taken is called shooting average.

The ratio of sides of a right triangle is determined by comparing the lengths of two sides.


To introduce ratios of the sides,
let's use the 3-4-5 right triangle again.

With each angle, there are only six possible ways that the sides can be divided into each other, so there are only six possible ratios for each angle. The six ratios for the reference angle shown (indicated by the shading) are as listed:

Each ratio has been assigned a name, and these names are called functions.

This chart shows the functions and their ratios.

is the Greek letter theta. It is a variable which is used to represent the degrees of an angle when the degrees are unknown.
In the chart above, is used to represent any angle.

For example, the sine function can be read: The sine of any angle is the length of the hypotenuse divided into the length of the opposite side.

This chart shows the functions and the ratios for the reference angle. Notice that the ratios are expressed as both fractions and decimals.

The hypotenuse is always the longest side. Notice that when the length of the hypotenuse is the numerator, the ratio is always greater than one. When the length of the hypotenuse is the denominator, the ratio is always less than one.

Here are the standard abbreviations for the functions.

Function practice: Determine the six functions for each angle of these right triangles. Round off to 4 decimal places.

Remember: There are two reference angles in each right triangle. The right angle is never the reference angle.


On to Function Memory Aid

[Privacy Policy] [Terms of Use]

Home || The Math Library || Quick Reference || Search || Help 

© 1994- The Math Forum at NCTM. All rights reserved.

Johnny & Margaret Hamilton
Please direct inquiries to main@constructpress.com
11 September 1995
Web page design by Sarah Seastone for the Geometry Forum