Calculating the Area of an Isosceles Triangle with a Perimeter of 8 cm

Given a triangle with integral sides and a perimeter of 8 cm, finding its area involves a combination of geometric principles and mathematical formulas. This article explores the process of determining a valid set of side lengths and then calculating the area using both manual and Heron's formula methods.

Introduction to the Problem

A triangle with integral sides and a perimeter of 8 cm needs to be analyzed. The task is to find the area of this triangle. First, we determine the possible combinations of side lengths that satisfy both the perimeter and the triangle inequality.

Identifying Possible Side Lengths

Let the sides of the triangle be a, b, and c. We have:

a b c 8

The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side:

a b c, a c b, b c a.

After exploring the possible combinations, the only valid set of side lengths is 2, 3, 3. This set satisfies the perimeter condition and the triangle inequalities.

Calculating the Area Using Heron's Formula

To calculate the area of a triangle with sides of 2 cm, 3 cm, and 3 cm, we first find the semi-perimeter s:

s a b c / 2 8 / 2 4.

Using Heron's formula, the area A is given by:

A √s(s-a)(s-b)(s-c)

Substituting the values:

a 2 cm, b 3 cm, c 3 cm, s 4 cm. A √(4(4-2)(4-3)(4-3)) √(4·2·1·1) √8 2√2 ≈ 2.83 cm2.

Thus, the area of the triangle is approximately 2.83 cm2.

Alternative Method Using Right Triangle

Another approach involves using the properties of a right triangle. If an isosceles triangle with sides 3 cm, 3 cm, and 2 cm is drawn, a perpendicular can be dropped from the vertex opposite the side of length 2 cm to the midpoint of that side. This forms a right triangle.

Let's call the midpoint D. Here, BD is 1 cm, AD is the height of the triangle, and AB (which is the same as AC) is 3 cm.

Using the Pythagorean theorem in the right triangle ABD:

AB2 AD2 BD2, 9 AD2 1, AD2 8, AD √8 2√2.

The area of the triangle can then be calculated as:

Area 1/2 × base × height 1/2 × 2 × 2√2 2√2 square cm.

Both methods yield the same result, confirming the area of the triangle.

Conclusion

The area of a triangle with integral sides and a perimeter of 8 cm, when determined using either the Heron's formula or the right triangle method, is approximately 2.83 cm2. This solution encapsulates the beauty of geometric and algebraic principles in solving real-world problems.