# Right triangle

see note below (*)
see note below (*)

New This calculator automatically draws the triangle to scale.

You may enter angles in different units like degree, percentage, radian and multiple of pi radian. To enter pi/6 angle, enter \alpha = 1/6 and choose from the units drop-down list "× pi radians".

## Right triangle formulas

### Pythagorean theorem

In a right triangle, the square of the hypotenus is equal to the sum of squares of the other two sides.

c^2 = a^2+b^2

The converse is also true. For a given triangle, if the square of the longest side is equal to the sum of squares of the other two sides then this triangle is right-angled.

### Trigonometric ratios in a right triangle

sin(\alpha) = b / c ;  cos(\alpha) = a / c

tan(\alpha) = b / a ;  cot(\alpha) = a / b

Less used ratios are secant (sec) and cosecant (csc).

sec(\alpha) = c / a = 1/cos(\alpha) ;  csc(\alpha) = c / b = 1/sin(\alpha)

In other words,

sine = opposite side / hypotenuse
cosine = adjacent side / hypotenuse
tangent = opposite side / adjacent side
cotangent = adjacent side / opposite side

And for less used ratios,
secant = hypotenuse / adjacent side
cosecant = hypotenuse / opposite side

### Complementary angles

In a right triangle, the acute angles \alpha and \beta are complementary because the sum of the three angles is 180 degrees and the third (right) angle is 90 degrees. So,

\alpha + \beta = 90°

### Perimeter and area

The perimeter of a triangle is simply equal to the sum of its three sides.
P = a+b+c

The area of the right triangle is equal to, A = (a*b)/2

### Height Calculation

#### Calculating the height h from the perpendicular sides a and b

h = (a*b)/c = (a*b)/sqrt(a^2+b^2)

This formula is based on the similarity of the triangles (h, q, b) and (a, b, c). Therefore,

h/b=a/c

#### Right triangle altitude theorem

h : altitude (or height) on the hypotenuse
p : projection of leg a on the hypotenuse
q : projection of leg b on the hypotenuse

The square of the altitude on the hypotenuse is equal to the geometric mean of the projections of the legs (non-hypotenuse sides) on the hypotenuse.

h^2 = p*q

Indeed, the triangles formed by the sides (h, q, b) and (p, h, a) are similar (since they have three equal angles), therefore,

h/q=p/h

### Leg geometric mean Theorem or Leg rule

The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the projection of that leg on the hypotenuse.

a^2 = p*c <=> a = sqrt(p*c)
b^2 = q*c <=> b = sqrt(q*c)