Height & Distance Concepts

Understanding the Concepts of Height & Distance

Trigonometric formulas may be used to calculate the height and distance of everything.

1. Calculation of Height Using Trigonometry

If you know the distance between your position and the building and the angle at which you can view the top, you can calculate the height.

We use trigonometry to calculate the height because the surface of the ground, the height of the building, and the line of elevation all form a right angle triangle with 90 degrees between the building and the ground.

Base

The 'base' of the right-angled triangle is created by the height of the building and line of sight is distance. The horizontal level's length also indicates the distance since it forms the base of the triangle, which is at a specific height but parallel to the ground.

Hypotenuse

The 'hypotenuse' of a right-angled triangle is the line of sight. The 'height' of the triangle is formed by a line perpendicular to the ground.

We can use the following formula to calculate the angle of elevation or depression:

$Sin 0=\frac{\mathrm{height}}{\mathrm{hypotenuse}} \cos 0=\frac{\mathrm{base}}{\mathrm{hypotenuse}} tan 0=\frac{\mathrm{height}}{\mathrm{base}}$

Here, θ is either the angle of elevation or depression.

Here AB is the base and BC is the height.

The angle of elevation

If the object is above the horizontal level, the angle of elevation is generated by the line of sight and the horizontal level.

The angle of depression

If the object is below the horizontal level, the Angle of Depression is the angle created between the line of sight and the horizontal level.

2. Pythagorean Theorem

Pythagoras' theorem states that the square of a right-angled triangle's hypotenuse equals the sum of the squares of its base and height.

Pythagoras' theorem can be used to calculate the length of the sides of a right-angled triangle when height and distance are involved.

${\left(\mathrm{hypotenuse}\right)}^2= {\left(\mathrm{base}\right)}^2 + {\left(\mathrm{height}\right)}^2$

If the length of the base, height and hypotenuse of a right-angle triangle is a, b and c respectively.

Then, $a^{2 + b^2 = c^2}$

Thus, if the length of any two sides is known then the length of the third side can be found by using the Pythagoras theorem which is also called the Pythagorean triplets.