What Are the Formulas for Heights and Distances?

Trigonometric ratios can be used to find heights and distances. Some useful relations are illustrated by the below-given diagrams which help us to determine heights and distances.

Challenging Question

1. The angles of depression of the top and the bottom of an 8yd8yd tall building from the top of a multi-storeyed building are 30∘30∘ and 45∘45∘, respectively. Find the height of the multi-storeyed building and the distance between the two buildings.Hint: Try to depict the figure according to a given situation.

How to Find Height and Distances?

To measure the heights and distances of different objects, we use trigonometric ratios. 

Use the Tangent rule to calculate the height of the tree (above eye level).
tan(angle) = opposite/adjacent
Where the opposite is the height of the tree and adjacent is the distance between you and the tree.

This is rearranged to:
opposite = tan(angle) x adjacent

or more simply
height=tan(angle)×distanceheight=tan(angle)×distance

Distance can be calculated as:

B (distance)=A (height)tan (e)B (distance)=A (height)tan (e)

Therefore, to calculate BB (distance) we will need the value of AA (height) and angle ee.