The traffic signal board is in the shape of an equilateral triangle. We need to find its area using Heron's formula. The perimeter of the triangle is given as 180 cm.
- The triangle is equilateral, so all sides are equal. - Perimeter of the triangle = 180 cm. - Let the side of the triangle be \( a \).
The perimeter \( P \) of an equilateral triangle is given by: \[ P = 3a \] Given \( P = 180 \, \text{cm} \), we can solve for \( a \): \[ 3a = 180 \quad \Rightarrow \quad a = \frac{180}{3} = 60 \, \text{cm} \]
For any triangle, the area \( A \) is given by Heron's formula: \[ A = \sqrt{s(s - a)(s - b)(s - c)} \] where \( s \) is the semi-perimeter and \( a, b, c \) are the sides of the triangle. Since the triangle is equilateral, \( a = b = c = 60 \, \text{cm} \). The semi-perimeter \( s \) is: \[ s = \frac{a + b + c}{2} = \frac{60 + 60 + 60}{2} = 90 \, \text{cm} \] Now, substituting the values into Heron’s formula: \[ A = \sqrt{90(90 - 60)(90 - 60)(90 - 60)} = \sqrt{90 \times 30 \times 30 \times 30} \] \[ A = \sqrt{90 \times 27000} = \sqrt{2430000} \] \[ A = 1560 \, \text{cm}^2 \]
The area of the signal board is \( \boxed{1560 \, \text{cm}^2} \).