# Formulas and Tables

Elementary Geometry# Equilateral Triangle

Angle of an equilateral triangle: \(\alpha = 60^\circ\)

Perimeter: \(P\)

Altitude: \(h\)

Radius of the circumscribed circle: \(R\)

Radius of the inscribed circle: \(r\)

Area: \(S\)

- In an equilateral triangle, the altitude, angle bisector, median and perpendicular bisector drawn from any vertex coincide.
- Relationship between the altitude (median, angle bisector or perpendiculr bisector) and the side

\(h = {\large\frac{{a\sqrt 3 }}{2}\normalsize}\) - Radius of the circumscribed circle (circumradius) of an equilateral triangle

\(R = {\large\frac{{2h}}{3}\normalsize} = {\large\frac{{a\sqrt 3 }}{3}\normalsize}\) - Radius of the inscribed circle (inradius) of an equilateral triangle

\(r = {\large\frac{{h}}{3}\normalsize} = {\large\frac{{a\sqrt 3 }}{6}\normalsize}\) - Relation between the circumradius and inradius in an equilateral triangle

\(R = 2r\) - Perimeter of an equilateral triangle

\(P = 3a = 6\sqrt 3 r =\) \(3\sqrt 3 R\) - Area of an equilateral triangle

\(S = {\large\frac{{ah}}{2}\normalsize} = {\large\frac{{{a^2}\sqrt 3 }}{4}\normalsize} =\) \({\large\frac{{3{R^2}\sqrt 3 }}{4}\normalsize} =\) \(3\sqrt 3 {r^2}\)