Learn: Easting increment
A survey line can be split into east and north coordinate changes. This lesson finds its east component from horizontal length and azimuth measured clockwise from north.
What the formula is saying
Convert azimuth θ from degrees to radians, then multiply L by sin θ. Because zero azimuth points north, the east component is zero there and uses sine.
Read the symbols in plain language
- L
- Line length
Line length. The direction fraction multiplied by plan length gives a signed displacement, not a final coordinate.
mMetres measure length; 1 m = 1000 mm.
- θ
- Bearing angle
Clockwise azimuth from grid north in this lesson: 0° north, 90° east. It is not a counterclockwise mathematical angle from the x axis.
degAngles are entered in degrees; multiply by π/180 for trigonometric calculations in radians.
- ΔE
- Result to find
Easting increment. The direction fraction multiplied by plan length gives a signed displacement, not a final coordinate.
m
Sort out the units first
L is the horizontal plan length in m; θ is azimuth in degrees from north, clockwise. The answer is a signed coordinate change in m: east is positive and west negative.
Assumptions before calculating
Assume a local planar coordinate system with known north direction and horizontal distance. Reduce slope distance to horizontal distance first and use one consistent grid or reference meridian.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find ΔE and explain the result in the stated output unit.
- L · Line length
- 100 m
- θ · Bearing angle
- 30 deg
Convert azimuth to radians
The calculator trigonometric function uses radians while survey azimuth is entered in degrees.
(30) × π ÷ 180 ≈ 0.5235987756 radResolve the direction component
Retain the signed trigonometric factor so the coordinate change points to the correct quadrant.
sin((0.5235987756)) = 0.5Scale the component by horizontal length
The direction fraction multiplied by plan length gives a signed displacement, not a final coordinate.
(100) × (0.5) = 50 m
Does this worked answer make sense?
At 0° the east change is zero; at 90° it equals +L. Together, ΔE² + ΔN² should equal L² apart from rounding.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- L · Line length
- 80 m
- θ · Bearing angle
- 210 deg
Convert azimuth to radians
The calculator trigonometric function uses radians while survey azimuth is entered in degrees.
(210) × π ÷ 180 ≈ 3.665191429 radResolve the direction component
Retain the signed trigonometric factor so the coordinate change points to the correct quadrant.
sin((3.665191429)) = -0.5Scale the component by horizontal length
The direction fraction multiplied by plan length gives a signed displacement, not a final coordinate.
(80) × (-0.5) = -40 m
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- L · Line length
- 120 m
- θ · Bearing angle
- 135 deg
Find: Learn: Easting increment
A hint, not the answer
Convert azimuth θ from degrees to radians, then multiply L by sin θ. Because zero azimuth points north, the east component is zero there and uses sine.
L is the horizontal plan length in m; θ is azimuth in degrees from north, clockwise. The answer is a signed coordinate change in m: east is positive and west negative.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Convert azimuth to radians
The calculator trigonometric function uses radians while survey azimuth is entered in degrees.
(135) × π ÷ 180 ≈ 2.35619449 radResolve the direction component
Retain the signed trigonometric factor so the coordinate change points to the correct quadrant.
sin((2.35619449)) ≈ 0.7071067812Scale the component by horizontal length
The direction fraction multiplied by plan length gives a signed displacement, not a final coordinate.
(120) × (0.7071067812) ≈ 84.85281374 m
Avoid the common trap
Do not use the mathematical convention of angle measured counterclockwise from east without conversion. Do not swap sine and cosine or take an absolute value that hides westward or southward motion.
When this method applies — and when it does not
This is a plane-survey component calculation, not a geodesic solution on the Earth. Grid convergence, scale factors, instrument errors and traverse adjustment are outside this formula. Add the change to a known start coordinate to obtain an endpoint.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
One idea understood. Keep going.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Easting increment. Elementary surveying, distances, angles, levels and areas. Keep a consistent coordinate system and distinguish a horizontal distance from a slope distance.
- Food and Agriculture Organization of the United Nations — Simple Methods for Aquaculture: Topography
- U.S. Army Corps of Engineers — Control and Topographic Surveying, EM 1110-1-1005 (2007)
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.