# Altitude of equilateral triangle

In this chapter we will derive the formula for altitude length for equilateral triangle.

## Formula for altitude length – equilateral triangle

The triangle in which all sides are of equal length are called equilateral triangle.

Given below is an equilateral triangle ABC with all sides measure ” a ” cm.

The line AM is the altitude of the triangle.

Let us find area of the triangle using Heron’s formula.

Calculating the semi perimeter (s) first.

\mathtt{s=\ \frac{a+a+a}{2}}\\\ \\ \mathtt{s=\ \frac{3}{2} a}

Now using the area formula;

\mathtt{Area\ =\sqrt{s( s-a)( s-b)( s-c)}}\\\ \\ \mathtt{Area\ =\ \sqrt{\frac{3}{2} a\ \left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)}}\\\ \\ \mathtt{Area\ =\ \sqrt{\frac{3}{2} a\times \left(\frac{a}{2}\right)\left(\frac{a}{2}\right)\left(\frac{a}{2}\right)}}\\\ \\ \mathtt{Area\ =\ \ \frac{\sqrt{3}}{4} a^{2}}

The area of equilateral triangle can also be calculated as;

\mathtt{Area\ =\ \frac{1}{2} \times base\times height}\\\ \\ \mathtt{Area\ =\ \frac{1}{2} \times a\times h}

Combining both the area equations.

\mathtt{\frac{1}{2} \times a\times h=\ \frac{\sqrt{3}}{4} a^{2}}\\\ \\ \mathtt{h\ =\ \frac{\sqrt{3}}{2} a}

Hence, height of equilateral triangle is given by formula \mathtt{\frac{\sqrt{3}}{2} a} .

Please remember the formula for examination purpose.

Let’s solve some problem related to the concept.

## Altitude of equilateral triangle – Solved Problems

Example 01
Given below is the equilateral triangle with sides 7 cm. Find the length of altitude of triangle.

Solution
The formula for altitude of equilateral triangle is given as;

\mathtt{h\ =\ \frac{\sqrt{3}}{2} a}

Putting the values, we get;

\mathtt{h\ =\ \frac{\sqrt{3}}{2} \times 7}\\\ \\ \mathtt{h\ =\ \frac{7\sqrt{3}}{2} \ cm}

Hence, \mathtt{\frac{7\sqrt{3}}{2} \ cm\ } is the length of altitude of triangle.

Example 02
In an equilateral triangle, if altitude length is \mathtt{5\sqrt{3}} then find the measure of each side of the triangle.

Solution
Altitude length formula for equilateral triangle is given as;

\mathtt{h\ =\ \frac{\sqrt{3}}{2} a}

Putting the values;

\mathtt{5\sqrt{3} \ =\ \frac{\sqrt{3}}{2} \times a}\\\ \\ \mathtt{a\ =\ \frac{5\sqrt{3}}{\sqrt{3}} \times 2}\\\ \\ \mathtt{a\ =\ 10\ cm}

Hence, each side of equilateral triangle measures 10 cm.