How to Calculate the Height of a Building Using Angles of Elevation - Lceted

How to Calculate the Height of a Building Using Angles of Elevation

One of the many ways this can be done is to view a building from two different ground points. Then the angles of elevation to the top of the building can be measured. The height and distance from the points can be worked out by using some very simple trigonometry. In this paper, we will consider an example involving such a problem and look at how one could make these calculations.

The Scenario

Consider the following case: Wanting to determine the height of a building, we have two points on the ground, at a known distance from one another, A and B, with which we measure angles of elevation to the top of a building. Here are the details for our example:

• Distance between points A and B (AB): 53 meters
• Angle of elevation from point A (α): 50 degrees
• Angle of elevation from point B (β): 65 degrees

Objective

1. The height of the building (h)
2. The distance from point B to the base of the building (BC)

Step-by-Step Solution

1. Set Up the Trigonometric Equations

Let:

• h = height of the building
• BC = distance from point B to the base of the building

From point A, we can write:

tan(α) = ℎ/BC

Therefore:

ℎ = BC ⋅ tan(α)

ℎ = BC ⋅ tan (50^∘)

From point B, we have:

tan(β) = h/(AB−BC)

Thus:

​ℎ = (AB−BC) ⋅ tan(β)

ℎ = (53−BC) ⋅ tan(65^∘)

2. Solve for the Distance 𝐵𝐶

Set the two expressions for ℎ equal:

BC ⋅ tan(50^∘) = (53−BC)⋅tan(65^∘)

Rearrange and solve for 𝐵𝐶

BC ⋅ tan(50^∘) = 53 ⋅ tan(65^∘) – BC ⋅ tan(65^∘)

BC ⋅ (tan(50^∘) + tan(65^∘)) = 53 ⋅ tan(65^∘)

BC = 53.tan(65^∘) / (tan(50^∘) + tan(65^∘)

Substitute the tangent values:

• tan(50^∘) - 1.1918
• tan(65^∘) - 2.1445

Thus:

BC - 53⋅2.1445​/1.1918+2.1445

BC – 113.66/3.3363

BC - 34.03 meters

3. Calculate the Height of the Building

Using the value of BC

ℎ = BC ⋅ tan(50^∘)

ℎ = 34.03⋅1.1918

ℎ = 40.58 meters

Summary

From two points of observation two angles of elevation can be measured and then the height of the building together with a distance from one point of observation to the base of a building is calculated. Example:

• Height of the building (h): Approximately 40.58 meters
• Distance from point B to the building (BC): Approximately 34.03 meters

This method is a practical application of trigonometry and can be particularly useful for surveying and construction projects where direct measurement is not feasible.

Conclusion

This is useful in various application areas: how to find out the height of a building using angles of elevation. With the correct measurements and a little trigonometric savvy, you will be able to deduce a pointedly accurate estimation of height and distance, thereby making it easier to plan and enabling the execution of the same.

Now, give this a try in your next survey project. And, as always, confirm that you have made your calculations correctly!