Understanding and Calculating the Height of an Equilateral Triangle (2023)

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. 1.812 cm
  2. 2.12 cm
  3. 2.598 cm
  4. 11 cm

Exercise 2

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

  1. 4.33 m
  2. 4.78 m
  3. 5.12 m
  4. 5.98 m

Exercise 3

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

  1. 6.1 cm
  2. 6.5 cm
  3. 7.1 cm
  4. 7.5 cm

Exercise 4

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

  1. 7.691 m
  2. 8.612 m
  3. 9.012 m
  4. 9.526 m

Exercise 5

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

  1. 10.39 cm
  2. 11.21 cm
  3. 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.

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