Finding the area of an equilateral triangle when you know its height is a straightforward process. This guide provides a clear, step-by-step approach, perfect for students and anyone needing a refresher on geometry. We'll explore the underlying principles and offer practical examples to solidify your understanding.
Understanding the Properties of an Equilateral Triangle
Before diving into the calculations, let's quickly review the key properties of an equilateral triangle:
- Equal Sides: All three sides are of equal length.
- Equal Angles: All three interior angles measure 60 degrees.
- Height Bisects Base: The height (or altitude) bisects the base, creating two congruent 30-60-90 right-angled triangles. This is crucial for our calculations.
Formula for the Area of a Triangle
The standard formula for the area of any triangle is:
Area = (1/2) * base * height
This formula is the foundation for solving our problem. However, we need to find the base length first since only the height is given.
Calculating the Base from the Height
Here's where the 30-60-90 triangle property comes into play. Consider one of the two right-angled triangles formed by the height. We know one angle (60 degrees) and the height (which is the side opposite the 60-degree angle).
We can use trigonometry (specifically, the tangent function) to find half the base:
tan(60°) = (height) / (half-base)
Solving for half the base:
half-base = height / tan(60°)
Since tan(60°) = √3, this simplifies to:
half-base = height / √3
To find the full base, simply double this value:
base = 2 * height / √3
Putting it all Together: Calculating the Area
Now that we have expressions for both the base and the height, substitute them into the area formula:
Area = (1/2) * base * height
Area = (1/2) * (2 * height / √3) * height
This simplifies to:
Area = (height² ) / √3
Or, rationalizing the denominator:
Area = (√3 * height²) / 3
Example: Finding the Area
Let's say the height of an equilateral triangle is 10 cm. Using our formula:
Area = (√3 * 10²) / 3 = (√3 * 100) / 3 ≈ 57.74 cm²
Therefore, the area of the equilateral triangle with a height of 10 cm is approximately 57.74 square centimeters.
Key Takeaways and Further Exploration
This step-by-step guide demonstrates how to efficiently calculate the area of an equilateral triangle using its height. Remember the crucial role of the 30-60-90 triangle and the trigonometric relationships involved. For further exploration, you might consider exploring other methods of calculating the area of an equilateral triangle, or investigating the relationship between area, height, and side length in different types of triangles. Understanding these concepts builds a solid foundation for more advanced geometric problems.
