Systematic corrections to a taped distance
A steel tape is only right at its standard temperature, pull, and support. Each departure has a formula, and each formula has a sign that the exam will test.
Temperature. Ct = 0.00000645 (T − 68 °F) L in feet, or 0.0000116 (T − 20 °C) L in metres. A hot tape has stretched, so each recorded 100 ft is really longer: the correction is positive.
Sag. A suspended tape hangs in a curve and reads long. Cs = −W² L / (24 P²), with W the total weight of the suspended span and P the pull. Always negative, and W is squared, which is the part people drop.
Pull. Cp = (P − P₀) L / (A E). More tension than standard stretches the tape; less lets it read long.
Slope. The horizontal distance is √(L² − h²). For gentle slopes the correction is h² / (2L), subtracted: 2.5 ft of rise over a 100 ft tape length costs 0.031 ft.
The sign rule
Think about what the tape physically does. A tape that is too long spans more ground per length, so you count fewer lengths and the recorded value is too small: add the correction when measuring an unknown distance. When laying out a known distance, the same tape needs the opposite treatment: subtract. Every sign trap in taping comes down to measuring versus laying out.
Normal tension
There is one pull at which the stretch from tension exactly cancels the shortening from sag, so a suspended tape reads its standardized length. It is defined by Pn = 0.204 W √(AE) / √(Pn − P₀), with Pn on both sides, so it is solved by iteration.