Triangle Area Calculation

The problem asks for the area of a triangle with a base of 8 cm and two equal sides of 6 cm. This is an isosceles triangle.

Finding Triangle Height

To calculate the area of a triangle, we use the formula:

\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)

We know the base is 8 cm, but we need to find the height. We can find the height by dividing the isosceles triangle into two identical right-angled triangles. The height acts as one leg, half the base (4 cm) is the other leg, and one of the equal sides (6 cm) is the hypotenuse.

Using the Pythagorean theorem (\(a^2 + b^2 = c^2\)):

\(\text{height}^2 + (\frac{\text{base}}{2})^2 = \text{side}^2\)

\(h^2 + (4 \text{ cm})^2 = (6 \text{ cm})^2\)

\(h^2 + 16 \text{ cm}^2 = 36 \text{ cm}^2\)

\(h^2 = 36 \text{ cm}^2 - 16 \text{ cm}^2\)

\(h^2 = 20 \text{ cm}^2\)

\(h = \sqrt{20 \text{ cm}^2} = \sqrt{4 \times 5} \text{ cm} = 2\sqrt{5} \text{ cm}\)

Calculating Triangle Area

Now we can calculate the area using the base and the calculated height:

\(\text{Area} = \frac{1}{2} \times 8 \text{ cm} \times 2\sqrt{5} \text{ cm}\)

\(\text{Area} = 4 \text{ cm} \times 2\sqrt{5} \text{ cm}\)

\(\text{Area} = 8\sqrt{5} \text{ cm}^2\)

The area of the triangle is \(8\sqrt{5}\) sq. cm.