Given: The sides of a triangle are in the ratio of 12:17:25 and the perimeter is 540 cm.

To find: The area of the triangle.

Solution:
To solve this problem, we can use the concept of the Heron's formula to find the area of the triangle.

Step 1: Identify the sides of the triangle
Let's assume the sides of the triangle are 12x, 17x, and 25x, where x is a common ratio.

Step 2: Calculate the perimeter of the triangle
The perimeter of a triangle is the sum of its three sides.
Perimeter = 12x + 17x + 25x = 54x

Given that the perimeter is 540 cm, we can equate it with the calculated perimeter to find the value of x.
54x = 540
Dividing both sides of the equation by 54, we get:
x = 10

So, the sides of the triangle are 120 cm, 170 cm, and 250 cm.

Step 3: Calculate the semi-perimeter
The semi-perimeter of a triangle is half of its perimeter.
Semi-perimeter = Perimeter / 2 = 540 / 2 = 270 cm

Step 4: Apply Heron's formula
Heron's formula states that the area of a triangle can be calculated using its semi-perimeter and the lengths of its sides.
Area = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter, a, b, and c are the lengths of the triangle's sides.

In our case, the lengths of the sides are 120 cm, 170 cm, and 250 cm, and the semi-perimeter is 270 cm.

Substituting the values into the formula, we get:
Area = √(270(270-120)(270-170)(270-250))
= √(270 * 150 * 100 * 20)
= √(729000000)
= 27000 cm²

Hence, the area of the triangle is 27000 cm².

Therefore, the area of the triangle with sides in the ratio of 12:17:25 and a perimeter of 540 cm is 27000 cm².