Levelling survey: finding heights with a level
A level is a small telescope on a tripod. It turns in a flat circle, so the line you see through it is perfectly level (horizontal).
- A staff is a tall ruler held upright on a point.
- Look through the level at the staff and read the number where the line cuts it. This is the reading.
- The first reading after setting up the level is the back sight (BS). The next is the fore sight (FS).
- Height difference = BS − FS. A positive answer is a rise (the front point is higher). A negative answer is a fall. A bigger reading means the point is lower.
- A point's height above a fixed zero level is its reduced level (RL).
Height of instrument (HI) method: HI = RL of known point + BS. Then RL of new point = HI − FS.
For a long line you move the level forward many times. A point where the staff stays while the level moves is a change point. At the end, check: ΣBS − ΣFS = last RL − first RL.
Longitudinal and cross-section surveying
A longitudinal section shows the ground height along the centre line of a road, canal or pipe. Put the staff at regular stations (for example every 10 m), read the heights, then plot distance along the page and height up the page.
Heights change much less than distances, so the drawing uses a bigger scale for height (for example 1 : 100) than for length (1 : 1000). This is called vertical exaggeration.
A cross-section is a slice at right angles to the centre line. It shows the ground across the width of the road. Readings are taken at the centre and a few metres left and right.
Together, the two sections give a 3D picture of the land.
Calculating volume and earthwork
Earthwork means digging and moving soil. Draw the planned final ground level (formation level) on the sections.
- Ground above the formation level is cut. It must be dug out.
- Ground below it is fill. Soil must be brought in.
- Area of a section = area between ground line and formation level.
- Average end area method: V = L × (A₁ + A₂) ÷ 2, where A₁ and A₂ are areas of two sections and L is the distance between them.
- Add up all pieces: total cut, total fill, then net = cut − fill. If cut is more, extra soil is carried away; if fill is more, soil is brought in.
Loose soil takes more space than packed ground (it "swells"), so truck loads are counted using a swell factor, for example 1.25.
Try it
In the 3D: go to the last step. Move the slider until cut and fill are almost equal. This is a balanced site where no soil needs to be brought in or taken out.
At home: hold a ruler upright on a table and on a book. Use a long straight plank resting on a stack of books as the "line of sight" and read how much lower the table is. Write BS and FS and find the difference.
Key formulas and definitions
- Height difference = BS − FS (positive = rise, front point is higher; negative = fall)
- HI = RL of known point + BS; RL of new point = HI − FS
- Check: ΣBS − ΣFS = last RL − first RL
- Volume between two sections: V = L × (A₁ + A₂) ÷ 2
- Net earthwork = total cut − total fill
- Key terms: level, staff, back sight, fore sight, reduced level, change point, section, cut, fill
Worked examples
1. Point A has RL 100.00 m. BS on A = 1.45 m, FS on B = 2.15 m. Find the RL of B.
HI = 100.00 + 1.45 = 101.45 m. RL of B = 101.45 − 2.15 = 99.30 m. B is 0.70 m lower than A.
2. A line has readings: BS 1.20, FS 1.60, BS 2.10, FS 1.30. The first RL is 50.00 m. Find the last RL by the check formula.
ΣBS = 3.30, ΣFS = 2.90. Difference = +0.40. Last RL = 50.00 + 0.40 = 50.40 m.
3. Two cross-sections 20 m apart have areas 12 m² and 18 m². Find the volume between them.
V = 20 × (12 + 18) ÷ 2 = 20 × 15 = 300 m³.
4. A road site has 450 m³ of cut and 380 m³ of fill. How much soil is left over or needed?
Net = 450 − 380 = 70 m³ more cut than fill. 70 m³ of soil is left over and must be carried away.
5. Cut of 300 m³ must be hauled. Swell factor 1.25, truck holds 6 m³ of loose soil. How many trips?
Loose soil = 300 × 1.25 = 375 m³. Trips = 375 ÷ 6 = 62.5, so 63 trips.
6. A canal bed must fall 0.5 m over 250 m. What is the slope? What RL does it reach if it starts at 80.00 m?
Slope = 0.5 ÷ 250 = 1 in 500 (0.002). End RL = 80.00 − 0.50 = 79.50 m.
Common mistakes
- Reading the staff upside down or leaning. Keep the staff upright; swing it slowly and read the smallest value.
- Thinking a bigger reading means a higher point. A bigger reading means the point is lower.
- Adding BS and FS. Always subtract: BS − FS.
- Mixing cut and fill. Ground above the formation level is cut; below is fill.