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 this geometry concept. We'll explore the underlying principles and formulas to ensure a thorough understanding.
Understanding Equilateral Triangles
Before diving into the calculations, let's refresh our understanding of equilateral triangles. An equilateral triangle is a triangle with all three sides of equal length. This property leads to several useful geometric relationships, including the fact that all its angles are equal to 60 degrees. This inherent symmetry simplifies area calculations significantly.
The Formula: Connecting Height and Area
The standard formula for the area of a triangle is:
Area = (1/2) * base * height
However, in an equilateral triangle, there's a direct relationship between the height and the base (which is also a side). This allows us to express the area solely in terms of the height.
Let's derive this relationship:
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Consider the 30-60-90 Triangle: Dropping a height from the apex of the equilateral triangle to the base bisects the base and creates two congruent 30-60-90 right-angled triangles.
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Trigonometric Relationships: In a 30-60-90 triangle, the ratio of the sides is 1:√3:2. If 'h' represents the height (opposite the 60° angle), and 'b' represents half the base, we have:
- b/h = 1/√3 => b = h/√3
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Finding the Base: Since 'b' is half the base, the full base ('B') is 2b = 2h/√3
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Substituting into the Area Formula: Now substitute the value of the base (B) into the general area formula:
Area = (1/2) * (2h/√3) * h = h² / √3
Step-by-Step Calculation
Let's say the height (h) of our equilateral triangle is 10 cm. Here's how to calculate the area:
Step 1: Square the height: 10² = 100 cm²
Step 2: Divide by the square root of 3: 100 cm² / √3 ≈ 57.74 cm²
Therefore, the area of the equilateral triangle with a height of 10 cm is approximately 57.74 cm².
Alternative Approach Using Heron's Formula (Less Efficient in this case)
While Heron's formula can calculate the area of any triangle given its side lengths, it's less efficient here. You'd first need to determine the side length from the given height using the 30-60-90 triangle relationship, adding extra steps. The method detailed above is more direct and simpler when the height is already known.
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