Equilateral triangles, characterized by equal side lengths, have a unique formula for calculating their height. In this article, we'll delve into the intricacies of determining the height using different methods, providing clarity on the mathematical principles involved.

## Formula for Height Calculation

The height ((h)) of an equilateral triangle with side length ((a)) can be found using the formula: [h = \frac{\sqrt{3}~a}{2}]

This formula, derived from the Pythagorean theorem, serves as the foundation for understanding the relationship between side length and height.

## Proof of the Height Formula

To comprehend the formula's origin, let's use the Pythagorean theorem ((a^2 = h^2 + \left(\frac{a}{2}\right)^2)) in the context of an equilateral triangle. Solving this equation yields: [h^2 = \frac{3a^2}{4}] [h = \frac{\sqrt{3}~a}{2}]

This rigorous proof establishes the reliability of the height formula.

## Calculating Height with Perimeter

If provided with the perimeter ((p)) of an equilateral triangle, determining the height involves first finding the side length ((a)). Given that (a = \frac{p}{3}), the height can be calculated using the standard formula.

[h = \frac{\sqrt{3}~p}{6}]

## Height Calculation with Area

When armed with the area ((A)), the process begins with obtaining the side length using the area formula: [A = \frac{\sqrt{3}}{4}~a^2]

Once (a) is determined, the height is easily calculated using the height formula.

## Resolving Height – Illustrated Examples

### Example 1

Given an equilateral triangle with sides of 2 m, the height ((h)) is calculated as follows: [h = \frac{\sqrt{3}~(2)}{2} = \sqrt{3}]

### Example 2

For a triangle with sides measuring 5 cm, the height ((h)) is determined as: [h = \frac{\sqrt{3}~(5)}{2} = 4.33]

### Example 3

If the height is known (6 cm), and the side length ((a)) is sought, the formula is rearranged to find (a): [a = \frac{12}{\sqrt{3}} \approx 6.928]

### Example 4

Given a height of 8 m, the corresponding side length ((a)) is found to be approximately 9.238 m.

### Example 5

With a perimeter of 30 cm, the height is calculated as: [h = \frac{\sqrt{3}~(10)}{2} \approx 8.66]

### Example 6

A triangle with a perimeter of 21 cm results in a height of approximately 5.196 cm.

### Example 7

For an equilateral triangle with an area of 60 cm², the height is determined as 10.193 cm.

## Practice Exercises

Now, let's apply this knowledge to some practice exercises.

### Exercise 1

If a right-angled triangle has sides of 3 cm, what is the height?

- 1.812 cm
- 2.12 cm
- 2.598 cm
- 11 cm

### Exercise 2

For an equilateral triangle with a perimeter of 15 m, what is the height?

- 4.33 m
- 4.78 m
- 5.12 m
- 5.98 m

### Exercise 3

Find the side length of an equilateral triangle with a height of 6.15 cm.

- 6.1 cm
- 6.5 cm
- 7.1 cm
- 7.5 cm

### Exercise 4

Determine the height of an equilateral triangle with sides measuring 11 m.

- 7.691 m
- 8.612 m
- 9.012 m
- 9.526 m

### Exercise 5

For a triangle with a perimeter of 36 cm, calculate the height.

- 10.39 cm
- 11.21 cm
- 11.92 cm

## Conclusion

Mastering the calculation of the height of an equilateral triangle allows for a profound understanding of its geometric properties. Whether approached through side length, perimeter, or area, the formula provides a versatile tool for solving various mathematical problems related to equilateral triangles.