An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

To find the area of an isosceles triangle, we need to know the length of the base. In this case, we have an isosceles triangle with two equal sides measuring 12 cm each. Let's denote the length of the base as 'b'.

The perimeter of a triangle is the sum of the lengths of its sides. In this case, we have two equal sides of 12 cm each, so the equation becomes:

12 cm + 12 cm + b = 30 cm

Simplifying the equation, we have:

24 cm + b = 30 cm

Subtracting 24 cm from both sides, we get:

b = 6 cm

Now we know that the length of the base is 6 cm. To find the area of the triangle, we can use the formula:

Area = (base * height) / 2

In an isosceles triangle, the height is the perpendicular distance from the base to the vertex opposite it. We can use the Pythagorean theorem to find the height.

Since the triangle is isosceles, we can draw an altitude from the vertex to the base, creating two congruent right triangles. Let's call the height 'h'. We have the following right triangle:


Using the Pythagorean theorem, we have:

h^2 + (b/2)^2 = 12^2

Substituting the values we know:

h^2 + (6/2)^2 = 12^2
h^2 + 3^2 = 144
h^2 + 9 = 144
h^2 = 144 - 9
h^2 = 135
h = √135
h ≈ 11.62

Now we have the height of the triangle, which is approximately 11.62 cm. Plugging this value into the area formula:

Area = (6 cm * 11.62 cm) / 2
Area ≈ 34.86 cm²

Therefore, the area of the isosceles triangle is approximately 34.86 square centimeters.