Calculating the Area of an Equilateral Triangle from its Perimeter

An equilateral triangle is a triangle where all three sides are equal in length, and all three angles are equal to 60 degrees.

The perimeter of any triangle is the sum of the lengths of its three sides. For an equilateral triangle, since all sides are equal, the perimeter is simply three times the length of one side.

Finding the Side Length of the Equilateral Triangle

Given the perimeter of the equilateral triangle is 48 cm.

Let 'a' be the length of each side of the equilateral triangle.

Perimeter = side + side + side = 3 \(\times\) side

Perimeter = \(3a\)

We are given: \(3a = 48\) cm

To find the side length 'a', we divide the perimeter by 3:

\(a = \frac{48}{3}\) cm

\(a = 16\) cm

So, the length of each side of the equilateral triangle is 16 cm.

Calculating the Area of the Equilateral Triangle

The formula for the area of an equilateral triangle with side length 'a' is:

Area = \(\frac{\sqrt{3}}{4} a^2\)

Now, we substitute the side length \(a = 16\) cm into the formula:

Area = \(\frac{\sqrt{3}}{4} \times (16)^2\)

Area = \(\frac{\sqrt{3}}{4} \times 256\)

Now, we can simplify the expression by dividing 256 by 4:

\(\frac{256}{4} = 64\)

So, the area is:

Area = \(64\sqrt{3}\) cm²

The area of the equilateral triangle with a perimeter of 48 cm is \(64\sqrt{3}\) cm².