1.16.7. Area and volumes

The land is always dealt with area. For any engineering project, land is acquired first on the basis of area requirement. In some projects, land may be required along a corridor of defined width (e.g., rail, road, pipeline etc.), while in other projects a large catchment area may be involved, such as flood protection devices, reservoir, etc. Thus, determining the requirement of area in a project is an essential part of surveying.

In engineering, the size of land needed for a project is essential. Engineers first measure how much area is necessary for the project. For example, if they're building a road, they wouldn't just need the road's width but the entire stretch of land along its path. Similarly, for projects like reservoirs or flood defenses, a large area of land is required to manage water flow effectively. Understanding the area helps ensure there is enough space to accommodate the proposed engineering works.

Imagine planning a picnic. Before you pick a spot, you first need to know how many friends are coming and what activities you want to do, which directly impacts the size of the area you choose. Similarly, engineers determine how much land they need based on the number of projects and their specific requirements.

It is very easy to determine the area of a regular figure (plot of land), such as a triangle, a square, a rectangle, etc. However, most of the time, the project land is irregular in shape with no defined shape, so computation of area is not straight forward in such cases. For this purpose, from a survey line offsets are taken at regular intervals, and area is calculated from any of the following methods: (i) Trapezoidal rule (ii) Simpson’s rule (iii) Using planimeter (iv) From coordinates.

Calculating area is straightforward for regular shapes like squares or rectangles since they have known formulas. However, real project sites are often irregularly shaped. To calculate their area, surveyors use techniques like taking measurements at intervals along a surveyed line, effectively breaking the land into manageable sections. Four methods can be employed: the trapezoidal rule uses trapezoids to approximate the area, Simpson’s rule uses parabolas for more accuracy, a planimeter is a tool that physically measures areas on maps, and coordinates method relies on geographic coordinates to calculate area of closed shapes.

Think about trying to cover your irregularly shaped backyard with sod. Instead of measuring the whole yard at once, you could measure small portions, calculate their areas, and then sum them up to know how much sod you need. This is similar to how surveyors deal with irregular land by breaking it into smaller, easier-to-handle sections.

In trapezoidal rule, the area is divided into a number of trapezoids, and boundaries are assumed to be straight between pairs of offsets. The area of each trapezoid is determined and added together to compute the entire area.

The trapezoidal rule approximates the area under a curve by dividing it into several trapezoids. By calculating the area of each trapezoid (the shape formed between two points), surveyors can approximate the total area of the irregular land. This method works by taking average heights at either end of the trapezoid and multiplying by the distance between them, and then summing up these areas for all trapezoids to get the total area.

Picture trying to find the area of a winding riverbank. Instead of measuring the whole bank, you think of it as a series of straight segments (like trapezoids) where you measure the length and height of each segment. By calculating the area of each segment and adding them up, you’ve effectively measured the total area of the riverbank without needing to measure its real, curvy edges.

In Simpson's rule, it is assumed that the irregular boundary is made up of parabolic arcs. The areas of the successive pairs of intercepts are added together to get the total area.

Simpson's rule provides a more accurate estimation for area by approximating the land boundary with parabolic arcs rather than straight lines. This means the irregular shape is represented more closely, and the area can be found by calculating the areas of these arcs and summing them up. It typically provides better accuracy than the trapezoidal rule, especially for curves and changes in elevation.

Imagine trying to cut a fruit like an orange in half to estimate how much juice it holds. Instead of making straight cuts, if you follow the natural curves of the fruit, you’ll get a more precise measurement of its volume. Utilizing curves (or parabolas in our case) helps to determine the total area of irregular shapes more accurately than using only straight lines.

Planimeter is a mechanical instrument used for measuring the area of a plan. It consists of a hard steel roller and a disc. The axis of roller coincides with the axis of tracer arm...

A planimeter is a device that physically traces the outline of an area to calculate its size. The roller moves as the user follows the perimeter of the irregular shape, creating a reading on the instrument that correlates to the area. This tool is especially useful for shapes that cannot be easily calculated mathematically. The area can be determined using a formula that takes initial and final readings along with the number of revolutions made by the roller.

Think of tracing the outline of your hand on a piece of paper to find out how much space it takes. When you’re done, you can measure around the outline. The planimeter is like the sophisticated version of tracing, providing a way to calculate the area precisely without needing to know the complicated math behind it.

The area of an irregular closed polygon is usually computed by the coordinate method. In this procedure, coordinates of each turning point in the figure must be known...

The coordinate method allows for area calculations of any closed shape by using the specific x and y coordinates of each angle or corner. The basic idea is to input these coordinates into a formula that combines them in a way to find the area of the polygon. This method can easily be executed by computer software, making it effective for complex shapes.

Imagine a treasure map with various landmarks marked in coordinates. If you know the coordinates of the corners of a treasure area, you can use these points to calculate the area. It’s like using GPS to determine how much land you have to dig in—by inputting the coordinates into a formula, you find out exactly how much area is covered.

The computation of volumes from the measurements taken in the field is required in design and planning of several engineering works...

Determining volume is crucial for projects like roads or reservoirs, where knowing how much material needs to be moved (like dirt or water) is necessary for planning. Surveyors typically take measurements at intervals across a specified line. Using results from cross-sectional views at these intervals, they apply the trapezoidal or prismoidal rules to calculate the total volume involved with earthworks.

Think of filling a backyard pool with water. To estimate how much water you need, you would measure its dimensions. For an irregular pool, you might break it down into smaller rectangular sections to find the volume of water needed to fill it up. Similarly, engineers take measurements and use specific rules to figure out the total materials needed for construction projects.

It is also known as average or mean sectional area formula. This method is based on the assumption that the mid-area of a pyramid is half the average area of the ends...

The trapezoidal rule for volume calculations works similarly to its area calculations. It assumes that the average area between the cross-sections can be used to find the total volume. By measuring the areas of the base sections, surveying teams can estimate the volume of earth they need to move or manipulate during construction projects.

Imagine a hawk browsing a field. If the hawk notices that each section of the field has different amounts of bait, it calculates the average amounts of bait it can gather. Similarly, engineers sum up the volumes of shapes made between known sections (like looking at the field in segments) to estimate the total volume of material present in a construction area.

The prismoidal formula can be used to compute volume of all geometric solids that are considered as prismoids. A prismoid, as shown in Figure 1.37, is a solid having ends that are parallel but not similar and trapezoidal sides...

The prismoidal rule is used for more complex shapes that are not uniform, like those seen in construction. It allows for the calculation of volume by considering not just the areas of the ends but also adjusting for the shape of the sides, which can vary or taper. This method delivers more accurate results for irregular and variable forms.

Think of making a cake in an irregular-shaped pan. The shape means the volume of the batter you need won’t be uniform; it may be deeper or shallower in certain spots. When engineers calculate how much material is needed for earthworks, they account for these irregularities similarly, ensuring they gather the right amount of earth or material for solid structures.

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