The Height of an Equilateral Triangle and Its Area Calculation

Understanding the relationship between the height and the side length of an equilateral triangle can be a powerful tool in geometry. In this article, we will discuss how to find the area of an equilateral triangle given its height and explore a variety of methods to solve a specific problem.

Introduction to Equilateral Triangles

An equilateral triangle is a triangle in which all three sides are equal and all internal angles are 60 degrees. Given the height of an equilateral triangle, we can determine the side length and subsequently, the area of the triangle. This article will guide you through a step-by-step process using multiple methods to arrive at the same result.

Using the Height to Find the Side Length

Let's consider an example where the height (h) of an equilateral triangle is 18 cm. The relationship between the height and the side length (s) of an equilateral triangle is given by:

[ h frac{sqrt{3}}{2} s ]

We are given that the height (h) is 18 cm:

[begin{align*} 18 frac{sqrt{3}}{2} s end{align*}]

By isolating 's', we get:

[begin{align*} s frac{2 times 18}{sqrt{3}} frac{36}{sqrt{3}} 12sqrt{3} text{ cm} end{align*}]

Calculating the Area Using the Formula

Now that we have the side length, we can use the formula for the area of an equilateral triangle:

[begin{align*} text{Area} frac{sqrt{3}}{4} s^2 frac{sqrt{3}}{4} times (12sqrt{3})^2 frac{sqrt{3}}{4} times 144 times 3 frac{sqrt{3}}{4} times 432 108sqrt{3} text{ cm}^2 end{align*}]

For an approximate numerical value:

[begin{align*} 108sqrt{3} approx 186.96 text{ cm}^2 end{align*}]

Alternative Methods to Calculate the Area

There are several other approaches to finding the area of an equilateral triangle when the height is given. These methods involve different formulas and step-wise calculations based on the same principle.

Method 1: Using Direct Calculation

Given that the height is 18 cm, we can use the direct relationship to find the side length:

[begin{align*} frac{sqrt{3}}{2} s 18 s frac{2 times 18}{sqrt{3}} frac{36}{sqrt{3}} 12sqrt{3} text{ cm} end{align*}]

Then, using the area formula:

[begin{align*} text{Area} frac{sqrt{3}}{4} times (12sqrt{3})^2 108sqrt{3} text{ cm}^2 end{align*}]

Method 2: Using Trigonometric Formulas

Another method uses trigonometric formulas. Given the height (h) and one side of the triangle (a), we can find the side length:

[begin{align*} sqrt{3} a / 2 18 a frac{2 times 15}{sqrt{3}} frac{30}{sqrt{3}} 10sqrt{3} text{ cm} end{align*}]

Then the area is calculated as:

[begin{align*} text{Area} frac{1}{2} times a times h frac{1}{2} times 10sqrt{3} times 15 75sqrt{3} text{ cm}^2 end{align*}]

Method 3: Using Cotangent and Trigonometric Formulas

Using the formula involving cotangent (cot 60° 1/√3), we can calculate the side length:

[begin{align*} a 2 times 18 cot 60 2 times 18 times frac{1}{sqrt{3}} 20.7846 text{ cm} text{Area} frac{18 times 20.7846}{2} 187.06 text{ cm}^2 end{align*}]

Method 4: Using Fundamental Geometry Principles

The height and side relationship can also be derived using the Pythagorean theorem:

[begin{align*} 18^2 left(frac{a}{2}right)^2 a^2 324 frac{a^2}{4} a^2 324 frac{3a^2}{4} a frac{60}{sqrt{3}} 20.7846 text{ cm} text{Area} frac{sqrt{3}}{4} times 20.7846^2 187.06 text{ cm}^2 end{align*}]

Conclusion

Whether you use the direct area formula, trigonometric relationships, or fundamental geometry principles, the result remains consistent. The area of the equilateral triangle with a height of 18 cm is approximately 187.06 cm2, or more precisely, 108√3 cm2. This article demonstrates the importance of understanding geometric relationships and the versatility of mathematical tools in solving geometric problems.

Keywords: equilateral triangle, height, area calculation