Formula

Area and altitude of an equilateral triangle

Mensuration • Plane Figures • Triangle

Formula

A = \frac{3}{4} a^{2}

An equilateral triangle of side

a

has altitude

\frac{3}{2} a

and area

\frac{3}{4} a^{2}

How to use it

1Reach for it whenever a figure decomposes into equilateral pieces: a regular hexagon of side $a$ is six of them, so its area is $\frac{3 3}{2} a^{2}$ .
2If the formula is forgotten, rebuild it in one line: the altitude bisects the base, so $h = a^{2} - (\frac{a}{2})^{2} = \frac{3}{2} a$ and the area is $\frac{1}{2} ah$ .
3The gap between three mutually touching circles of radius $r$ is this area with $a = 2 r$ , less half a circle.

Find the altitude and the area of an equilateral triangle of side

6

cm.

Step-by-step

Answer

altitude

33

cm, area

93

^{2}

Watch out

The coefficient

\frac{3}{4}

multiplies

a^{2}

, not

a

, and it is easily confused with the altitude's

\frac{3}{2}

— the two differ by exactly the factor

\frac{a}{2}

of the half-base-times-height.

Builds on

Also known as

area of an equilateral triangle, root 3 over 4 a squared, altitude of an equilateral triangle, equilateral triangle height