The Compendium Geometry is an eBook providing facts, formulas and explanations about geometry.


Home Analytic Geometry Triangle Properties of a Right Triangle
See also: Triangle - General Definitions, Properties of Arbitrary Triangles, Altitudes of a Triangle, Isosceles Triangle, Pythagorean Theorem, Thales' Theorem, Incircle and Angle Bisectors of a Triangle, Median and Centroid of a Triangle, Ellipse 

Properties of a Right Triangle

A right triangle has one angle (the angle g at the point C by convention) of 90 degrees (p/2). The longest side, which is opposite to the angle g is called hypothenuse (the word derives from the Greek hypo - "under" - and teinein - "to stretch"). The legs of a right triangle (i.e. the sides adjacent to the right angle) are also known as catheti (singular: cathetus). In German, the two legs are denoted using different terms: Ankathete (adjacent cathetus) and Gegenkathete (opposite cathetus) denote the legs adjacent to and opposite the (non-right) angle in question, respectively.

The sides of the triangle are related to each other by the trigonometric functions:

sin = opposite / hypothenuse
sin a = a / c
sin b = b / c

cos = adjacent / hypothenuse
cos a = b / c
cos b = a / c

tan = opposite / adjacent
tan a = a / b
tan b = b / a

Hint: A helpful mnemotic to remember the above relationships is

"Tommy On A Ship Of His Caught A Herring"

(T...tangens, S...sine, C...cosine, A...adjacent, O...opposite, H...hypothenuse).

Further the altitude of the triangle h is given by

h = ab/c,

the radius of the incircle is defined by

ri = (a + b - c)/2,

the radius of the circumcircle is given by

rcc = c/2, and

the center of gravity is at one third of the height h.

If two parameters of a right triangle are known, all other parameters can be calculated. The following table contains the most important parameters (three sides a, b, c, two angles a and b and the area).

Known properties Properties to calculate Area A
a, b a = arctan(a/b) b = arctan(b/a) ab/2
a, c a = arcsin(a/c) b = arccos(a/c)
b, c a = arccos(b/c) b = arcsin(b/c)
a, a b = acot(a) c = acsc(a) b = p/2 - a a2/2cot(a)
a, b b = atan(b) c = asec(b) a = p/2 - b a2/2tan(b)
b, a a = btan(a) c = bsec(a) b = p/2 - a b2/2tan(a)
b, b a = bcot(b) c = bcsc(b) a = p/2 - b b2/2cot(b)
c, a a = csin(a) b = ccos(a) b = p/2 - a c2/4sin(2a)
c, b a = ccos(b) b = csin(b) a = p/2 - b c2/4sin(2b)

Last Update: 2007-Sep-03