Calculation of Area using Trapezoidal Formula

Given Data:

Distance in m: 0, 5, 10, 15, 20, 30, 40

Offsets in m: 3, 4, 5.5, 5, 6, 4, 4.5

Boundary offsets are taken from the survey line.

Step 1: Plotting the Data

The first step is to plot the data on a graph with distance on the x-axis and offset on the y-axis. We can use Microsoft Excel or any other plotting software to do this.

Step 2: Calculation of Area

The trapezoidal rule is a numerical integration method used to approximate the definite integral of a function. In this case, we can use the trapezoidal rule to calculate the area between the survey line and the boundary.

The formula for the trapezoidal rule is:

Area = (b1 + b2)h/2

where b1 and b2 are the lengths of the parallel sides of the trapezoid and h is the height of the trapezoid.

To apply this formula to our data, we need to divide the area into trapezoids. We can do this by drawing vertical lines at each survey point and then connecting the points with straight lines.

For example, the first trapezoid has a height of 3 meters and parallel sides of 0 meters and 5 meters. The area of this trapezoid is:

Area1 = (0 + 5) x 3/2 = 7.5 square meters

Similarly, we can calculate the area of all the trapezoids and add them up to get the total area between the survey line and the boundary.

Step 3: Calculation of Total Area

After calculating the area of all the trapezoids, we can add them up to get the total area between the survey line and the boundary.

Total Area = Area1 + Area2 + ... + AreaN

where N is the number of trapezoids.

Conclusion:

In this way, we can calculate the area between the survey line and the boundary using the trapezoidal formula. This method is widely used in civil engineering for calculating the area of irregular shapes.