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.
Answer
The height is given by $h = \frac{\sqrt{3}}{2} s$.
Answer for screen readers
The height of an equilateral triangle with side length $s$ is given by:
$$ h = \frac{\sqrt{3}}{2} s $$
Steps to Solve
- 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 $s$.
- 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.
- 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:
$$ h = \frac{\sqrt{3}}{2} s $$
More Information
The formula for the height of an equilateral triangle is derived from the geometric properties and ratios of a special right triangle (30-60-90 triangle). It's useful not only for theoretical math but also in practical applications such as architecture and design.
Tips
- Confusing the height with other properties such as the area. Remember, height is the perpendicular distance from the base to the opposite vertex.
- Miscalculating or misapplying the properties of a 30-60-90 triangle; always double-check the ratios.