# The altitude of an equilateral triangle is 18 inches. What is the length of a side?

May 14, 2016

The side  color(blue)( = 12sqrt3  inches.

#### Explanation:

The altitude of the triangle = color(blue)( 18 inches

Let us denote the side of the triangle as $a$.

The formula for calculating the altitude of an equilateral triangle is :

Altitude $= \frac{\sqrt{3}}{2} \times a$ (side)

$18 = \frac{\sqrt{3}}{2} \times a$

$a = 18 \times \frac{2}{\sqrt{3}}$

$a = \frac{36}{\sqrt{3}}$

 a = (36 xx color(blue)(sqrt3)) / (sqrt3 xx color(blue)(sqrt3)

$a = \frac{36 \sqrt{3}}{3}$

$a = 12 \sqrt{3}$ inches.

May 14, 2016

Length of the side of the triangle is $12 \sqrt{3}$ inches.

#### Explanation:

If the length of the side of a triangle is $a$ and $h$ be its height as shown below.

As the perpendicular from vertex divides base equally, we can find height $h$ using Pythagoras theorem

Hence $h = \sqrt{{a}^{2} - {\left(\frac{a}{2}\right)}^{2}} = \sqrt{{a}^{2} - {a}^{2} / 4} = \sqrt{3 {a}^{2} / 4} = \frac{\sqrt{3}}{2} a$

But as height is $18$ inches, $\frac{\sqrt{3}}{2} a = 18$ or

$a = 18 \times \frac{2}{\sqrt{3}} = 36 \frac{\sqrt{3}}{3} = 12 \sqrt{3}$