Formulas and Tables

Elementary Geometry

Isosceles Triangle

Base: \(a\)
Legs: \(b\)
Base angle: \(\beta\)
Vertex angle: \(\alpha\)

Altitude to the base: \(h\)
Perimeter of an isosceles triangle: \(P\)
Area of an isosceles triangle: \(S\)

  1. An isosceles triangle is a triangle that has two equal sides. The equal sides are called the legs and the third side is called the base. In the figure below, the legs and the base are denoted by the letters \(b\) and \(a,\) respectively.
  1. Relationship between the vertex and base angles
    \(\beta = 90^\circ – {\large\frac{\alpha }{2}\normalsize}\)
  2. Altitude drawn to the base
    \({h^2} = {b^2} – {\large\frac{{{a^2}}}{4}\normalsize}\)
  3. In an isosceles triangle, the altitude, angle bisector, median and perpendicular bisector drawn from the vertex to the base coincide.
  4. Relationships between the legs and the base
    \(b = 2a\cos \alpha,\;\) \(b = 2a\sin {\large\frac{\beta }{2}\normalsize}\)
  5. Perimeter of an isosceles triangle
    \(P = a + 2b\)
  6. Area of an isosceles triangle
    \(S = {\large\frac{{ah}}{2}\normalsize} =\) \({\large\frac{{{b^2}}}{2}\normalsize}\sin \alpha =\) \({\large\frac{{ab}}{2}\normalsize}\sin \beta \)