1.20.1. Finding height of an object which is accessible

Let PP’ is a tower whose elevation of the top is to be determined. Set up the theodolite at a convenient ground point A so that the top of tower and a staff kept on the BM can be bisected. Measure vertical angle α of the top of tower as well as take the staff reading at the BM. Measure D, the horizontal distance between the theodolite station and tower.

In this step, we are preparing to measure the height of the tower (PP'). First, we need to set up a theodolite at a location, denoted as point A. The theodolite is a precise instrument used for measuring angles in vertical and horizontal planes. The goal in this setup is to line up the theodolite's telescope so that it can see the top of the tower and a staff (measuring rod) positioned on the benchmark (BM) below it. Next, we need to measure the vertical angle (α) which is the angle formed by the line of sight to the top of the tower and the horizontal line of the theodolite. We also take the horizontal distance (D) from the theodolite to the base of the tower, which will be needed to calculate the height later.

Imagine you want to measure the height of a tall building using a protractor (like the theodolite). You stand a certain distance away from the building and aim your protractor to look at the top of the building, while also marking your height with a measuring stick at a reference point on the ground. This helps you determine how tall the building is by combining the angle you see and your distance from it.

To find the height of the object above a BM: Let H = height of the object above the BM, h = height of the tower above the instrument axis, h s = height of the instrument axis above the BM, s α = vertical angle of the top of tower at the instrument station, D = horizontal distance from the instrument station to the base of the tower. Then, h = D tan α H = h + h s = D tan α + h s.

In this step, we use the measurements we've taken to calculate the height of the tower (H) above the benchmark. The first step is to calculate 'h', which is the height from the instrument (theodolite) to the top of the tower. This is calculated using the formula h = D tan α, where D is the horizontal distance and α is the vertical angle we measured earlier. Once we have 'h', we can find 'H', the total height of the tower above the benchmark by adding 'h' (the height of the tower above the instrument) and 'h s' (the height of the instrument axis above the benchmark). Thus, H = D tan α + h s. This gives us the total height of the tower.

Think of trying to figure out how tall a ladder is when it is propped against a wall. You stand a certain distance from the wall (horizontal distance, D) and look up at the top of the ladder (which forms the angle α with the ground). By using trigonometry (using the tangent), you can calculate how tall the ladder reaches before it touches the wall and add your own height above the ground (h s) to find the total height above the ground.

If the distance D is large, correction for curvature and refraction, i.e., 0.00673(D^2/1000) is to be applied.

In some cases, especially when the distance D is large, we must account for the effect of the Earth's curvature and atmospheric refraction. This is because as the distance increases, these factors can lead to a slight inaccuracy in our measurements. The formula used for this correction is 0.00673(D^2/1000), which identifies how much more we need to add to our height calculation to account for these distortions. It's important to make this correction to ensure that our final height measurements are as accurate as possible.

Imagine trying to measure the distance across a large lake. As you look across, the water may look curved rather than straight, especially if you are focusing on a point far away. Similarly, when measuring heights over long distances, we have to consider that the Earth is round and that our line of sight may not follow a perfect straight line due to this curvature.

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