Distance Measurement by Sine Law

If we need to specify a certain distance to a point or an object without the need to go to that object, then the following formula may help. Tools needed are only measuring angles tool or protractor or compass, and measuring distances tool or measuring tape. 

By using the sine formula on a triangle as follows:

a / sin a = b / sin b = L / sin g

Where a and b is not necessarily equal or should not an isosceles triangle. And g can be determined as follow: 

180^o – a – b – g = 0^o

g = 180^o – a – b

Picture a triangle above is considered as a top view of a plan to measure the distance from A to objects or measure the distance from B to object.

If the angle a is known, the distance between A to B or L is known, and the angle b is known, then the distance between A to object, and the distance between B to object can be calculated, without the need to go to the object directly.

The distance between A to object is:
B = sin b x L / sin b g = sin x L / sin (180 - a - b)

The distance between B to object is:
a = sin a x L / sin a g = sin x L / sin (180 - a - b)

If the distance between A to B or L is 50 meter, the angle is 55^o and angle a b is 60^o then:
Distance from A to object = b = sin 60 x 50 / Sin (180-55 -60) = 47.8 meters
Distance from B to object = a = sin 55 x 50 / Sin (180-55 -60) = 45.2 meters

Results with the best accuracy obtained if a and b are about  35 to 55 degrees. By using a Theodolite or Total Station equipment as used by the surveyors in the field of geodesy, the measurement can be performed with accuracy up to seconds.

Common angle measurements are in degrees, minutes, and seconds (DMS). One degree is equal to 60 minutes, and one minute is equal to 60 seconds. One rotation is 360 degrees.

This Excel sheet is containing the following formula to calculate the distance to an object.

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

For angle measurement, always use the degrees unit.