Calculating the Area of an Equilateral Triangle

The problem asks us to find the area of an equilateral triangle when its side length is given. An equilateral triangle is a triangle where all three sides are equal in length, and all three angles are equal (each being 60 degrees).

Understanding the Formula for Equilateral Triangle Area

The area of an equilateral triangle can be calculated using a specific formula that depends only on the length of its side. If the side length of the equilateral triangle is denoted by \(s\), the area \(A\) is given by the formula:

\[A = \frac{\sqrt{3}}{4} \times s^2\]

Applying the Formula to Find the Area

We are given that the side length of the equilateral triangle is 8 cm. Let's plug this value into the formula:

\[s = 8 \text{ cm}\]

Substitute \(s = 8\) into the area formula:

\[A = \frac{\sqrt{3}}{4} \times (8 \text{ cm})^2\]

First, calculate the square of the side length:

\[8^2 = 8 \times 8 = 64\]

Now substitute this back into the formula:

\[A = \frac{\sqrt{3}}{4} \times 64 \text{ cm}^2\]

Next, simplify the expression:

\[A = \sqrt{3} \times \frac{64}{4} \text{ cm}^2\]

Divide 64 by 4:

\[\frac{64}{4} = 16\]

So, the area of the equilateral triangle is:

\[A = 16\sqrt{3} \text{ cm}^2\]

Comparing with the Options

Let's look at the given options and compare them with our calculated area:

• Option 1: \(32\sqrt{3} \text{ cm}^2\)
• Option 2: \(8\sqrt{3} \text{ cm}^2\)
• Option 3: \(64\sqrt{3} \text{ cm}^2\)
• Option 4: \(16\sqrt{3} \text{ cm}^2\)

Our calculated area, \(16\sqrt{3} \text{ cm}^2\), matches Option 4.

Conclusion on Equilateral Triangle Area

The area of the equilateral triangle with a side length of 8 cm is \(16\sqrt{3}\) cm². This calculation used the standard formula for the area of an equilateral triangle based on its side length.

Revision Table: Equilateral Triangle Properties

Additional Information on Triangles

Triangles are fundamental shapes in geometry. Here are some key concepts related to triangles:

• Types of Triangles: Triangles can be classified by their side lengths (scalene, isosceles, equilateral) or by their angles (acute, right, obtuse).
• Area of Any Triangle: The general formula for the area of any triangle is \(A = \frac{1}{2} \times \text{base} \times \text{height}\). For an equilateral triangle, the height can be derived in terms of the side, leading to the specific formula used above.
• Pythagorean Theorem: For right-angled triangles, the square of the hypotenuse is equal to the sum of the squares of the other two sides (\(a^2 + b^2 = c^2\)). This theorem is often used in deriving formulas for triangle properties.
• Units: When calculating area, the units are always squared (e.g., cm², m²). Perimeter is measured in linear units (e.g., cm, m).