Finding the Area of an Equilateral Triangle with One Side Length

An equilateral triangle is a unique geometric shape where all three sides are equal in length. This simplicity allows us to deduce many interesting properties, including the formula to calculate its area. In this article, we will explore how to determine the area of an equilateral triangle when the length of one side is known. Particularly, we will use the example where one side of the triangle is 4 cm.

Area Formula for Equilateral Triangles

The area of an equilateral triangle can be calculated using the formula:

Area (frac{sqrt{3}}{4} times s^2)

where s represents the length of a side of the triangle.

Example Calculation

Let's consider an equilateral triangle where one side is 4 cm. Using the formula stated above, the area can be determined as follows:

Area (frac{sqrt{3}}{4} times 4^2)

Breaking it down:

4^2 16 (frac{sqrt{3}}{4} times 16 4sqrt{3}) cm2

Thus, the area of the equilateral triangle is approximately:

4(sqrt{3}) ≈ 6.93 cm2

Detailed Explanation and Additional Formulation

Let's delve deeper into the formula (frac{sqrt{3}}{4} times s^2). Here, the multiplier (frac{sqrt{3}}{4}) is derived from the geometry of the equilateral triangle, which ensures that the calculation gives precise results.

A step-by-step breakdown of the calculation is:

Pow the side length s by 2: (s^2) Multiply the result by (sqrt{3}): (sqrt{3} times s^2) Divide the product by 4: (frac{sqrt{3}}{4} times s^2)

Alternative Derivation Using Half-Base and Height

Another way to derive the area of an equilateral triangle is by using the half-base and height. In an equilateral triangle, the height can be calculated using the Pythagorean theorem:

Height (sqrt{s^2 - left(frac{s}{2}right)^2} sqrt{s^2 - frac{s^2}{4}} sqrt{frac{3s^2}{4}} frac{sqrt{3}}{2} times s)

Given that the base is equal to the side length s, the area can also be calculated as:

Area (frac{1}{2} times s times frac{sqrt{3}}{2} times s frac{sqrt{3}}{4} times s^2)

Conclusion

In summary, the area of an equilateral triangle with a side length of 4 cm is approximately 6.93 square centimeters. This calculation is based on the general formula (frac{sqrt{3}}{4} times s^2), where (s) is the side length of the triangle. The formula can be derived from geometric properties and is applicable to any equilateral triangle, regardless of its side length.