Formulas and Tables

Elementary Geometry

Right Triangle

Legs of a right triangle: \(a\), \(b\)
Hypotenuse of a right triangle: \(c\)
Acute angles: \(\alpha\), \(\beta\)
Right angle: \(C\)
Area: \(S\)

Altitude to the hypotenuse: \(h\)
Medians: \({m_a}\), \({m_b}\), \({m_c}\)
Circumradius: \(R\)
Inradius: \(r\)

  1. A right triangle is a triangle in which one angle is a right angle (equal to \(90^\circ\)).
  2. The side opposite of the right angle is called the hypotenuse. The sides adjacent to the right angle are called the legs. In the figure below, \(AC\) and \(BC\) are the legs, \(AB\) is the hypotenuse. The lengths of the legs are \(a\), \(b\). The length of the hypotenuse is \(c\).
A right triangle
  1. The sum of the acute angles of a right triangle is \(90^\circ\):
    \(\alpha + \beta = 90^\circ\)
  2. The sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:
    \(\sin \alpha = {\large\frac{a}{c}\normalsize},\;\) \(\sin \beta = {\large\frac{b}{c}\normalsize}\)
  3. The cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse:
    \(\cos \alpha = {\large\frac{b}{c}\normalsize},\;\) \(\cos \beta = {\large\frac{a}{c}\normalsize}\)
  4. The tangent of an acute angle is the ratio of the opposite leg to the adjacent leg:
    \(\tan \alpha = {\large\frac{a}{b}\normalsize},\;\) \(\tan \beta = {\large\frac{b}{a}\normalsize}\)
  5. The cotangent of an acute angle is the ratio of the adjacent leg to the opposite leg:
    \(\cot \alpha = {\large\frac{b}{a}\normalsize},\;\) \(\cot \beta = {\large\frac{a}{b}\normalsize}\)
  6. The secant of an acute angle is the ratio of the hypotenuse to the adjacent leg:
    \(\sec \alpha = {\large\frac{c}{b}\normalsize},\;\) \(\sec \beta = {\large\frac{c}{a}\normalsize}\)
  7. The cosecant of an acute angle is the ratio of the hypotenuse to the opposite leg:
    \(\csc \alpha = {\large\frac{c}{a}\normalsize},\;\) \(\csc \beta = {\large\frac{c}{b}\normalsize}\)
  8. Pythagorean theorem
    The sum of the squares of the legs is equal to the square of the hypotenuse:
    \({a^2} + {b^2} = {c^2}\)
  9. \({a^2} = fc,\;\) \({b^2} = gc,\)
    where \(f\) and \(g\) are projections of the legs \(a\) and \(b,\) respectively, onto the hypotenuse \(c\).
  1. \({h^2} = fg\),
    where \(h\) is the altitude drawn to the hypotenuse \(c\), and \(f\) and \(g\) are projections of the legs \(a\) and \(b\) onto the hypotenuse.
  2. Medians drawn to the legs of a right triangle
    \(m_a^2 = {b^2} – {\large\frac{{{a^2}}}{4}\normalsize},\;\) \(m_b^2 = {a^2} – {\large\frac{{{b^2}}}{4}\normalsize},\)
    where \({m_a}\) and \({m_b}\) are the medians drawn to the legs \(a\) and \(b,\) respectively.
  3. Median drawn to the hypotenuse of a right triangle
    \({m_c} = {\large\frac{c}{2}\normalsize}\,\), where \({m_c}\) is the median to the hypotenuse \(c.\)
Median drawn to the hypotenuse of a right triangle
  1. Radius of the circumcircle of a right triangle
    \(R = {\large\frac{c}{2}\normalsize} = {m_c}\)
  2. Radius of the incircle of a right triangle
    \(r = {\large\frac{{a + b – c}}{2}\normalsize} =\) \({\large\frac{{ab}}{{a + b + c}}\normalsize}\)
  3. \(ab = ch\)
  4. Area of a right triangle
    \(S = {\large\frac{{ab}}{2}\normalsize} = {\large\frac{{ch}}{2}\normalsize}\)