A polygon traverse has the following coordinates shown below. Calculate the area within the traverse using the cross coordinate method.
A polygon traverse has the following coordinates shown below. Calculate the area within the traverse using the cross coordinate method.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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To calculate the area enclosed by this polygon using the cross coordinate method, follow these steps:
1. **List the Coordinates**:
Ensure that the coordinates are listed in a consistent order (clockwise or counterclockwise).
2. **Apply the Cross Product Formula**:
Use the cross product formula to calculate the area:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i \cdot y_{i+1} - y_i \cdot x_{i+1}) + (x_n \cdot y_1 - y_n \cdot x_1) \right|
\]
Where \((x_i, y_i)\) represents the coordinates of the \(i^{th}\) vertex and \(n\) is the number of vertices.
3. **Substitute the Coordinates**:
Plug in the given coordinates into the formula.
4. **Calculate the Determinants**:
Compute the determinant for each coordinate pair and sum them up.
#### Example Calculation
Given the coordinates:
\[
\begin{array}{cc}
x & y \\
\hline
625.255 & 511.356 \\
970.356 & 540.356 \\
980.564 & 400.552 \\
928.562 & 220.322 \\
650.395 & 255.852 \\
\end{array}
\]
The area calculation will involve these cross products:
\[
\frac{1}{2} \left| 625.255 \cdot 540.356 + 970.356 \cdot 400.552 + 980.564 \cdot 220.322 + 928.562 \cdot 255.852 + 650.395 \cdot 511.356 \right.
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Transcribed Image Text:### Polygon Traverse Area Calculation Using Cross Coordinate Method
Consider a polygon traverse with the following coordinates:
- \( (625.255, 511.356) \)
- \( (970.356, 540.356) \)
- \( (980.564, 400.552) \)
- \( (928.562, 220.322) \)
- \( (650.395, 255.852) \)
The diagram below depicts a plotted polygon based on these coordinates:
To calculate the area enclosed by this polygon using the cross coordinate method, follow these steps:
1. **List the Coordinates**:
Ensure that the coordinates are listed in a consistent order (clockwise or counterclockwise).
2. **Apply the Cross Product Formula**:
Use the cross product formula to calculate the area:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i \cdot y_{i+1} - y_i \cdot x_{i+1}) + (x_n \cdot y_1 - y_n \cdot x_1) \right|
\]
Where \((x_i, y_i)\) represents the coordinates of the \(i^{th}\) vertex and \(n\) is the number of vertices.
3. **Substitute the Coordinates**:
Plug in the given coordinates into the formula.
4. **Calculate the Determinants**:
Compute the determinant for each coordinate pair and sum them up.
#### Example Calculation
Given the coordinates:
\[
\begin{array}{cc}
x & y \\
\hline
625.255 & 511.356 \\
970.356 & 540.356 \\
980.564 & 400.552 \\
928.562 & 220.322 \\
650.395 & 255.852 \\
\end{array}
\]
The area calculation will involve these cross products:
\[
\frac{1}{2} \left| 625.255 \cdot 540.356 + 970.356 \cdot 400.552 + 980.564 \cdot 220.322 + 928.562 \cdot 255.852 + 650.395 \cdot 511.356 \right.
\]
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