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

Properties of a Right Triangle

A right triangle has one angle (the angle γ at the point C by convention) of 90 degrees (π/2). The longest side, which is opposite to the angle γ 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 α = a / c
sin β = b / c

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

tan = opposite / adjacent
tan α = a / b
tan β = 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 α and β and the area).

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

Last Update: 2011-01-11