If three sides of the triangle is given 18 cm, 41 cm and 41 cm. Then the area…

Concept

For a triangle whose three side lengths are known, the area is given by Heron's formula. If the sides are a, b, c and the semi-perimeter is s = (a + b + c)/2, then Area = √[ s(s − a)(s − b)(s − c) ]. For an isosceles triangle (two equal sides) the same result also follows from Area = ½ × base × height, where the height drops from the apex to the midpoint of the unequal base.

Application

1. Identify the sides: a = 18, b = 41, c = 41. The two equal sides are 41, so the base is 18.

2. Semi-perimeter: s = (18 + 41 + 41) / 2 = 100 / 2 = 50.

3. Substitute into Heron's formula: Area = √[ 50 × (50 − 18) × (50 − 41) × (50 − 41) ] = √[ 50 × 32 × 9 × 9 ].

4. Simplify under the root: 50 × 32 = 1600, and 9 × 9 = 81, so Area = √(1600 × 81) = √1600 × √81 = 40 × 9 = 360.

Therefore the area is 360 cm2.

Cross-check

Using the height method: the base 18 is split into two halves of 9. The height h satisfies h = √(41² − 9²) = √(1681 − 81) = √1600 = 40. Then Area = ½ × 18 × 40 = 360 cm2, which agrees with Heron's formula.