UNDERSTAND IT. WORK IT OUT.

Learn: Northing increment

A survey line can be split into east and north coordinate changes. This lesson finds its north component from horizontal length and azimuth measured clockwise from north.

Beginner-friendlyFree · No accountTwo worked examples + separate practice
01

What the formula is saying

Convert azimuth θ from degrees to radians, then multiply L by cos θ. Because zero azimuth points north, the north component equals the whole length there and uses cosine.

ΔN = L cosθ

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.

m

Metres 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.

deg

Angles are entered in degrees; multiply by π/180 for trigonometric calculations in radians.

ΔN
Result to find

Northing 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: north is positive and south 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.

02

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 ΔN and explain the result in the stated output unit.

L · Line length
100 m
θ · Bearing angle
30 deg

  1. Convert azimuth to radians

    The calculator trigonometric function uses radians while survey azimuth is entered in degrees.

    (30) × π ÷ 180 ≈ 0.5235987756 rad

  2. Resolve the direction component

    Retain the signed trigonometric factor so the coordinate change points to the correct quadrant.

    cos((0.5235987756)) ≈ 0.8660254038

  3. Scale the component by horizontal length

    The direction fraction multiplied by plan length gives a signed displacement, not a final coordinate.

    (100) × (0.8660254038) ≈ 86.60254038 m
Answer86.60254038 m

Does this worked answer make sense?

At 0° the north change equals +L; at 90° it is zero. 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

  1. Convert azimuth to radians

    The calculator trigonometric function uses radians while survey azimuth is entered in degrees.

    (210) × π ÷ 180 ≈ 3.665191429 rad

  2. Resolve the direction component

    Retain the signed trigonometric factor so the coordinate change points to the correct quadrant.

    cos((3.665191429)) ≈ -0.8660254038

  3. Scale the component by horizontal length

    The direction fraction multiplied by plan length gives a signed displacement, not a final coordinate.

    (80) × (-0.8660254038) ≈ -69.2820323 m
Answer-69.2820323 m
03

Now try your own values

Change a value or its unit. The same method will show your calculation, step by step.

Line length. The direction fraction multiplied by plan length gives a signed displacement, not a final coordinate.

Clockwise azimuth from grid north in this lesson: 0° north, 90° east. It is not a counterclockwise mathematical angle from the x axis.

English, Arabic and Persian digits are supported. The steps convert inputs to the formula’s base units.

Results update only when you calculate. The lesson example above stays unchanged.

04

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: Northing increment

For repeating decimals, use at least four significant figures. Accepted rounding tolerance: 0.05% of the expected value; zero uses an absolute tolerance of 10⁻¹².

A hint, not the answer

Convert azimuth θ from degrees to radians, then multiply L by cos θ. Because zero azimuth points north, the north component equals the whole length there and uses cosine.

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: north is positive and south negative.

Show the full practice solution

Compare the steps with your work; revealing a solution does not mark the lesson complete.

  1. Convert azimuth to radians

    The calculator trigonometric function uses radians while survey azimuth is entered in degrees.

    (135) × π ÷ 180 ≈ 2.35619449 rad

  2. Resolve the direction component

    Retain the signed trigonometric factor so the coordinate change points to the correct quadrant.

    cos((2.35619449)) ≈ -0.7071067812

  3. Scale 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
Answer-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.

This optional checkmark is saved only in this browser. It is your own progress note, not a certificate.

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: Northing increment. Elementary surveying, distances, angles, levels and areas. Keep a consistent coordinate system and distinguish a horizontal distance from a slope distance.

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.

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