You can determine the height of a building using trigonometry by measuring angles and distances from a specific vantage point.

To calculate the height of a building, start by measuring the horizontal distance from the base of the building to a point where the top of the building is clearly visible. This distance is referred to as the “adjacent” side in trigonometric terminology. Next, measure the angle of elevation from your observation point to the top of the building. The angle of elevation is defined as the angle between your line of sight to the top of the building and the horizontal ground.

With these two measurements—the distance and the angle of elevation—you can apply the tangent function, which is a fundamental trigonometric function. In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side (which represents the height of the building) to the length of the adjacent side (the distance you measured). This relationship can be expressed with the formula:

To find the height of the building, you can rearrange this formula as follows:

For instance, suppose you measure a distance of $50$ metres from the base of the building, and the angle of elevation to the top of the building is $30$ degrees. You would compute the height using the formula:

Using a calculator, you find that $\tan(30^\circ) \approx 0.577$. Therefore, the calculation proceeds as follows:

Thus, the height of the building is approximately $28.85$ metres.