What you need

  • The ground (natural surface) at the chainage: offset and R.L. of each surveyed point, from the reduced level book. Offsets left of the centre line are negative.
  • The design section: here a formation 10 m wide at R.L. 100.000, from offset −5.0 to +5.0.
  • The side slope from each edge of the formation down or up to the ground: here 1 vertical to 2 horizontal (1 V : 2 H) on both sides.
Offset (m) −12.0 −8.0 −4.0 0.0 4.0 8.0 12.0
Ground R.L. 100.550 100.400 100.050 99.700 99.850 100.250 100.500

Step 1: find the toe points

From each edge of the formation, run the side slope out until it meets the ground. If the ground at the edge is above the formation, the slope goes up (cut); if it is below, the slope goes down (fill).

Left edge (−5.0): the ground there, between −8.0 and −4.0, is 100.050 + ¼ × 0.350 = 100.138, above 100.000, so the slope rises outwards: level = 100.000 + (−5.0 − offset) ÷ 2. Setting that equal to the ground line between −8.0 and −4.0 gives offset −5.333, R.L. 100.167.

Right edge (+5.0): the ground there is 99.850 + ¼ × 0.400 = 99.950, below 100.000, so the slope falls outwards: level = 100.000 − (offset − 5.0) ÷ 2. It meets the ground line between 4.0 and 8.0 at offset +5.083, R.L. 99.958.

The design line is now: toe (−5.333, 100.167), edge (−5.0, 100.000), edge (+5.0, 100.000), toe (+5.083, 99.958).

Step 2: the depth at every offset

Take every offset where either line has a point, between the two toes, and work out d = ground − design at each. A positive d is cut, a negative d is fill.

Offset Ground Design d
−5.333 (toe) 100.167 100.167 0.000
−5.000 100.138 100.000 +0.138
−4.000 100.050 100.000 +0.050
0.000 99.700 100.000 −0.300
4.000 99.850 100.000 −0.150
5.000 99.950 100.000 −0.050
+5.083 (toe) 99.958 99.958 0.000

Step 3: split where ground and design cross

Between −4.000 (d = +0.050) and 0.000 (d = −0.300) the sign changes: the ground crosses the formation. The crossing is where d = 0:

offset = −4.000 + 4.000 × 0.050 ÷ (0.050 + 0.300) = −3.429

Add it as a point with d = 0, so no strip mixes cut and fill.

Step 4: add up the strips

Each pair of neighbouring offsets makes a trapezoid: area = (d₁ + d₂) ÷ 2 × width.

From To Width d₁ d₂ Area (m²) Type
−5.333 −5.000 0.333 0.000 0.138 0.023 cut
−5.000 −4.000 1.000 0.138 0.050 0.094 cut
−4.000 −3.429 0.571 0.050 0.000 0.014 cut
−3.429 0.000 3.429 0.000 −0.300 0.514 fill
0.000 4.000 4.000 −0.300 −0.150 0.900 fill
4.000 5.000 1.000 −0.150 −0.050 0.100 fill
5.000 5.083 0.083 −0.050 0.000 0.002 fill

Cut = 0.131 m², fill = 1.516 m².

(These figures come from the same calculation CivilSurvey Expert uses, run on the numbers above.)

Things to watch

  • Use every point of both lines. Leaving out a ground point between two design points cuts a corner off the area.
  • Always split at crossings. Averaging a positive and a negative depth in one strip makes cut and fill cancel out, and both come out too small.
  • Find the toes on the ground line, not by a fixed width. On sloping ground the two toes are at different distances from the centre line.
  • One layer at a time. A canal or road section has several layers (bed and banks, earth top and sub-base); each is measured against the layer below it the same way.

The next step is to turn section areas into volumes: see earthwork volume by the average end area method.