Window Area Calculation: Step-by-Step Solution

This problem asks us to find the total area of a unique window design. The window is made up of two distinct geometric shapes: a square portion and an equilateral triangle portion placed on top of it. We are given the total perimeter of this window, which is 6 meters, and our goal is to calculate its area in square meters.

Window Components and Properties

Let's first visualize and understand the structure of the window:

• The bottom part of the window is a square. A square has four equal sides and four right angles.
• Above the square is an equilateral triangle. An equilateral triangle has all three sides equal in length and all three angles equal to 60 degrees.
• A key detail is that the base of the triangular portion perfectly coincides with the upper side of the square. This means that the side length of the square is exactly the same as the side length of the equilateral triangle.

Determining the Side Length of the Window

To find the area, we first need to determine the side length of the square and the triangle. Let's denote this common side length as \(s\) meters.

The perimeter of the window is the total length of its outer boundary. Let's count how many sides of length \(s\) contribute to the perimeter:

• From the square portion: The bottom side, the left side, and the right side are part of the window's outer perimeter. The top side of the square is internal as it's shared with the triangle. So, the square contributes \(3\) sides of length \(s\).
• From the equilateral triangle portion: The two slanted sides of the triangle are part of the window's outer perimeter. The base of the triangle is internal as it coincides with the top side of the square. So, the equilateral triangle contributes \(2\) sides of length \(s\).

Therefore, the total perimeter \(P\) of the window is the sum of these external sides:

\[P = (\text{number of square sides in perimeter} \times s) + (\text{number of triangle sides in perimeter} \times s)\]

\[P = (3 \times s) + (2 \times s)\]

\[P = 3s + 2s\]

\[P = 5s\]

We are given that the perimeter of the window is 6 meters:

\[5s = 6 \text{ m}\]

Now, we can solve for \(s\):

\[s = \frac{6}{5} \text{ m}\]

\[s = 1.2 \text{ m}\]

So, the side length of the square and each side of the equilateral triangle is 1.2 meters.

Calculating the Area of Each Section

Now that we know the side length \(s = 1.2\) m, we can calculate the area of the square portion and the area of the equilateral triangle portion separately.

Square Portion Area

The formula for the area of a square is \( \text{side} \times \text{side} \), or \( \text{side}^2 \).

\[\text{Area}_{\text{square}} = s^2\]

Substitute \(s = 1.2\) m into the formula:

\[\text{Area}_{\text{square}} = (1.2 \text{ m})^2\]

\[\text{Area}_{\text{square}} = 1.44 \text{ m}^2\]

Equilateral Triangle Portion Area

The formula for the area of an equilateral triangle with side length \(s\) is:

\[\text{Area}_{\text{triangle}} = \frac{\sqrt{3}}{4} s^2\]

Substitute \(s = 1.2\) m into the formula:

\[\text{Area}_{\text{triangle}} = \frac{\sqrt{3}}{4} (1.2 \text{ m})^2\]

\[\text{Area}_{\text{triangle}} = \frac{\sqrt{3}}{4} (1.44 \text{ m}^2)\]

Using the approximate value of \( \sqrt{3} \approx 1.732 \):

\[\text{Area}_{\text{triangle}} \approx \frac{1.732}{4} \times 1.44 \text{ m}^2\]

\[\text{Area}_{\text{triangle}} \approx 0.433 \times 1.44 \text{ m}^2\]

\[\text{Area}_{\text{triangle}} \approx 0.62352 \text{ m}^2\]

Total Area of the Window

To find the total area of the window, we add the area of the square portion and the area of the equilateral triangle portion:

\[\text{Total Area}_{\text{window}} = \text{Area}_{\text{square}} + \text{Area}_{\text{triangle}}\]

\[\text{Total Area}_{\text{window}} = 1.44 \text{ m}^2 + 0.62352 \text{ m}^2\]

\[\text{Total Area}_{\text{window}} = 2.06352 \text{ m}^2\]

Rounding this value to two decimal places, we get approximately 2.06 m\(^2\).

Summary of Calculations

The area of the window is approximately 2.06 m\(^2\).