A window is made up of a square portion and an equilateral triangle portion above it. The base of the triangular portion coincides with the upper side of the square. If the perimeter of the window is 6 m, the area of the window in m2 is ___________.
2.06
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
| Description | Value |
|---|---|
| Given Perimeter of Window | 6 m |
| Perimeter Formula (\(P = 5s\)) | \(5s = 6\) |
| Calculated Side Length (\(s\)) | 1.2 m |
| Area of Square (\(s^2\)) | \( (1.2)^2 = 1.44 \text{ m}^2 \) |
| Area of Equilateral Triangle (\(\frac{\sqrt{3}}{4}s^2\)) | \( \frac{\sqrt{3}}{4}(1.2)^2 \approx 0.62352 \text{ m}^2 \) |
| Total Area of Window | \( 1.44 + 0.62352 \approx 2.06352 \text{ m}^2 \) |
The area of the window is approximately 2.06 m\(^2\).
Important Questions from Numerical Computation
Ratio of the angles of a triangle are 1 : 2 : 6. What is the smallest angle?
It would take one machine 4 hours to complete a production order and another machine 2 hour to complete the same order. If both machines work simultaneously at their respective constant rates, the time taken to complete the same order is ________ hours.
Two design consultants, P and Q, started working from 8 AM for a client. The client budgeted a total of USD 3000 for the consultants. P stopped working when the hour hand moved by 210 degrees on the clock. Q stopped working when the hour hand moved by 240 degrees. P took two tea breaks of 15 minutes each during her shift, but took no lunch break. Q took only one lunch break for 20 minutes, but no tea breaks. The market rate for consultants is USD 200 per hour and breaks are not paid. After paying the consultants, the client shall have USD_remaining in the budget.
What is the value of \(1 + \frac{1}{4} + \frac{1}{{16}} + \frac{1}{{64}} + \frac{1}{{256}} + \ldots ?\)
A 1.5 m tall person is standing at a distance of 3 m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters?
