What is the height of an equilateral triangle?

Understand the Problem

The question is asking for the formula or method to calculate the height of an equilateral triangle. We use the properties of equilateral triangles to derive the height from the side length.

The height is given by $h = \frac{\sqrt{3}}{2} s$.

The height of an equilateral triangle with side length $s$ is given by:

$$h = \frac{\sqrt{3}}{2} s$$

Steps to Solve

1. Identify the properties of an equilateral triangle

An equilateral triangle has all three sides equal and all three angles equal to $60^\circ$. Let the length of a side be denoted as $s$.

1. Use the Pythagorean theorem

To find the height, we'll split the equilateral triangle into two 30-60-90 triangles. The height will be the longer leg of this right triangle.

1. Find the height using the relationship in a 30-60-90 triangle

In a 30-60-90 triangle, the ratios of the sides are $1:\sqrt{3}:2$. The height (longer leg) is given by:

$$h = \frac{\sqrt{3}}{2} \cdot s$$

This comes from the fact that the height corresponds to the side opposite the $60^\circ$ angle.

The height of an equilateral triangle with side length $s$ is given by:

$$h = \frac{\sqrt{3}}{2} s$$