Measuring Height by Sine Formula

By simply using a protractor and measure the horizontal distance, you can calculate the height of a building, tower, tree, hill, even a mountain from a certain distance. Without the need to physically measure the height of those big objects.

This method uses a sine formula :

a / sin a = b / sin b = L / sin g

An altitude or height is the distance from the horizontal plane to a certain point higher or above it. In mathematics a height is always perpendicular to a horizontal plane. So for the sine formula above, b will always 90^o. In a triangle, the sum of all angles is always 180^o , if the angle a can be measured, then the angle g can be calculated. So the formula for measuring the object height is:

a = sin a x L / sin a g = sin x L / sin (90 - a)

a = calculated height
a = angle from the horizontal plane to the top of the object
L = horizontal distance between the observer to the foot of the object

Unit of measurement of distance and height that is calculated can be anything, such as: meters, km, feet, miles, etc.. 

For angle measurement, always use the degrees unit.

So, simply by measuring the distance from the observer to the base of a high object, and measuring the angle between the horizontal plane to an imagenary line to the top of the object, then the height of the object can be calculated.

Best calculation if a is about 35 to 55 degree.

This Excel spreadsheet is containing the formula to calculate object height.