In the construction of railway or highway, the natural earth surface does not usually present suitable formation level for such construction due to undulations. To bring the surfaces of the earth to a uniform formation level, there is the need to remove areas that are too high-**CUT** in order to bring them to the required level and fill areas that are too low-**FILL** in order to bring them to the required level.

Most times, the practice is to take the cut materials to fill up the fill areas if the material is good. This helps to save a lot of cost involved in bringing in new materials. To know whether there is need to bring in more materials to site or remove materials from the site and to know the volume expected here, it is necessary to compute the volume of cut and fill.

Generally, if (**Ʃ volume of cut – Ʃ volume of fill)** is positive, it means that some materials would be removed from the site. If the expression above gives a negative value, it means that some materials need to be brought into the site to make up level. But if the expression is zero, it means there is no need for haul in or haul out of materials.

Volume computation presently are integrated into many highway design software like Civil 3D, Civil CAD etc. and the volumes are computed alongside the design. Some other tools like Tekla Tedds can be used manually to determine this volume when the existing ground levels and formation levels are known (see attached PDF for cut and fill tutorial/example from Tekla Tedds software).

Tekla Tedds software can be obtained from HERE or HERE

**To obtain volume of cut or fill using freehand mathematical methods,**

Establish control pegs at regular intervals over the area. The pegs must have been carefully levelled from a bench mark or temporary benchmark within or close to site.

**Note;**

i. To fill an area, it is better that the pegs are placed so that their tops are the required level.

ii. To cut an area, sight rails is more suitable to be used.

In the classroom, the normal volume computation can be done using three (3) popular methods:

- By means of cross-sectional area multiplied by the length of the earthwork
- From contours
- From surface spot levels.

**Method of Cross-sectional Area**

In this method, the sections are taken at suitable intervals and the area of each section obtained. With the area of each cross-section and distance of successive sections, the total volume may be calculated by one of the following three methods:

**Mean Areas Method**

Here, the volume of section, V = (A_{1} + A_{2} + A_{3} +……+ A_{n}) x L

Where A_{1}, A_{2}, A_{3}, …An are cross-sectional areas, n is the number of cross-sections and L is the length of the cross-sectional area.

**End Areas Method or Trapezoidal Method**

In this method, volume, V = L_{1} ((A_{1}+A_{2})/2 ) + L_{2} ((A_{2}+A_{3})/2 ) + L_{3} ((A_{3}+A_{4})/2) +….+ L_{n-1} ((A_{n-1}+A_{n})/2)

Where, L = L_{1} = L_{2} = L_{3} = ……. = L_{n} = length of each trapezoid.

Note: If L = L_{1} = L_{2} = L_{3} = ……. = L_{n }are regular or equal intervals, then the formula can be written as V = L [(((A_{1}+A_{n})/2) + A_{2} + A_{3} + …….+ A_{n-1}].

**Simpson’s Rule or Prismoid Method**

Here, the rule states that the area enclosed by a curvilinear figure divided into an even number of strips of equal width is equal to one-third (1/3) of the width of a strip multiplied by the sum of the two extreme ordinates plus twice the sum of the remaining odd ordinates plus 4 times the sum of the even ordinates.

Mathematically,

Volume, V = L/3 (M + 2ƩO + 4ƩE)

Where, L = width of the strip

M = sum of the first and last strips

ƩO = sum of the odd ordinates

ƩE = sum of the even ordinates

It is believed that method 3 is more accurate than others

**Volume from contours**

In this method, the planimeter (see Figure 1) is used to measure the areas of irregular figures which are based on areas created by contour intervals. As we may well know, contours which are lines connecting areas of equal height or levels are usually irregular due to the irregularity of earth’s surface. Thus, areas created by contour intervals would be irregular and planimeter is used to measure the areas accurately.

Thereafter, the end areas method can be used to calculate the volume with formula below,

Volume, V = D/2 (A_{1} + A_{n} + 2 (A_{2} + A_{3} + …….+ A_{n-1})], where D is the contour interval.

**Volume from Spot heights**

Spot heights indicate the elevation at strategic points of a road section or building section. See the post: Use of spot height………

Thanks for reading!