Tape correction (surveying)

In surveying, tape correction(s) refer(s) to correcting measurements for the effect of slope angle, expansion or contraction due to temperature, and the tape's sag, which varies with the applied tension. Not correcting for these effects gives rise to systematic errors, i.e. effects which act in a predictable manner and therefore can be corrected by mathematical methods.
Correction due to slope.
formula_1
Where 
 L= Inclined length measured
 A= Inclined angle
When distances are measured along the slope, the equivalent horizontal distance may be determined by applying a slope correction.
The vertical slope angle of the length measured must be measured. (Refer to the figure on the other side) Thus,
Where:
The correction formula_9 is subtracted from formula_12 to obtain the equivalent horizontal distance on the slope line:
Correction due to temperature.
When measuring or laying out distances, the standard temperature of the tape and the temperature of the tape at time of measurement are usually different. A difference in temperature will cause the tape to lengthen or shorten, so the measurement taken will not be exactly correct. A correction can be applied to the measured length to obtain the correct length.
The correction of the tape length due to change in temperature is given by:
Where:
The correction formula_15 is added to formula_19 to obtain the corrected distance:
For common tape measurements, the tape used is a steel tape with coefficient of thermal expansion C equal to 0.000,011,6 units per unit length per degree Celsius change. This means that the tape changes length by 1.16 mm per 10 m tape per 10 °C change from the standard temperature of the tape. For a 30 meter long tape with standard temperature of 20 °C used at 40 °C, the change in length is 7 mm over the length of the tape.
Correction due to sag.
A tape not supported along its length will sag and form a catenary between end supports. The correction due to sag must be calculated separately for each unsupported stretch separately and is given by:
Evaluated at L=0, it vanishes and so does not contribute a Taylor series term. The second derivative of formula_22 is
Again, when evaluated at L=0 it vanishes. When evaluated at L=0, the third derivative survives, however.