Window Area Calculation: Perimeter and Composite Shape

The problem requires finding the area of a window composed of a square and an equilateral triangle.

Perimeter Determination

Let 's' represent the side length of the square. Since the equilateral triangle sits atop the square, sharing a side, the triangle's side length is also 's'.

The window's perimeter comprises the bottom edge of the square and the two vertical sides of the square, plus the two exposed vertical sides of the triangle.

• Bottom side of the square: s
• Left side (square + triangle): s + s = 2s
• Right side (square + triangle): s + s = 2s

Wait, let's reconsider the perimeter. The perimeter is the outer boundary. This includes:

• The bottom side of the square (length s).
• The left side of the square (length s).
• The right side of the square (length s).
• The left side of the triangle (length s).
• The right side of the triangle (length s).

Therefore, the total perimeter is the sum of these 5 sides:

Perimeter = s + s + s + s + s = 5s.

We are given that the perimeter is 6 m:

$5s = 6 \text{ m}$

Solving for the side length 's':

$s = \frac{6}{5} \text{ m} = 1.2 \text{ m}$

Area Calculation Steps

The total area of the window is the sum of the area of the square portion and the area of the equilateral triangle portion.

Square Area Calculation

The formula for the area of a square is $A_{square} = s^2$.

Using $s = 1.2$ m:

$A_{square} = (1.2 \text{ m})^2 = 1.44 \text{ m}^2$

Equilateral Triangle Area Calculation

The formula for the area of an equilateral triangle is $A_{triangle} = \frac{\sqrt{3}}{4} s^2$.

Using $s = 1.2$ m:

$A_{triangle} = \frac{\sqrt{3}}{4} (1.2 \text{ m})^2 = \frac{\sqrt{3}}{4} \times 1.44 \text{ m}^2$

$A_{triangle} = \sqrt{3} \times 0.36 \text{ m}^2$

Using the approximation $\sqrt{3} \approx 1.732$:

$A_{triangle} \approx 1.732 \times 0.36 \text{ m}^2 \approx 0.6235 \text{ m}^2$

Total Window Area

Sum the areas of the square and the triangle:

$A_{total} = A_{square} + A_{triangle}$

$A_{total} \approx 1.44 \text{ m}^2 + 0.6235 \text{ m}^2$

$A_{total} \approx 2.0635 \text{ m}^2$

Rounding to two decimal places gives the final area.

The area of the window is approximately 2.06 $m^2$.