# Equilateral Triangle

• Side of an equilateral triangle: $$a$$
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: $$S$$
1. An equilateral triangle is a triangle in which all three sides are equal. All angles in an equilateral triangle are equal to $$60^\circ$$.
2. In an equilateral triangle, the altitude, angle bisector, median and perpendicular bisector drawn from any vertex coincide.
3. Relationship between the altitude (median, angle bisector or perpendiculr bisector) and the side
$$h = {\large\frac{{a\sqrt 3 }}{2}\normalsize}$$
4. Radius of the circumscribed circle (circumradius) of an equilateral triangle
$$R = {\large\frac{{2h}}{3}\normalsize} = {\large\frac{{a\sqrt 3 }}{3}\normalsize}$$
5. Radius of the inscribed circle (inradius) of an equilateral triangle
$$r = {\large\frac{{h}}{3}\normalsize} = {\large\frac{{a\sqrt 3 }}{6}\normalsize}$$
6. Relation between the circumradius and inradius in an equilateral triangle
$$R = 2r$$
7. Perimeter of an equilateral triangle
$$P = 3a = 6\sqrt 3 r =$$ $$3\sqrt 3 R$$
8. Area of an equilateral triangle
$$S = {\large\frac{{ah}}{2}\normalsize} = {\large\frac{{{a^2}\sqrt 3 }}{4}\normalsize} =$$ $${\large\frac{{3{R^2}\sqrt 3 }}{4}\normalsize} =$$ $$3\sqrt 3 {r^2}$$