Lab 4: Area and Volume Computations

CE 111 Section 1

February 23, 2015

Crew:

Jordan Eikamp

Brandon Raleigh

Andrew Suiter

Objective:

The objective of this lab is to learn how to measure areas of irregular shapes using a

planimeter. Also, a goal of this lab is to determine areas and volumes (of cut and fill) of cross

sections from established coordinates.

Introduction:

Measuring areas of regions is an important part of surveying. Rather than physically

measuring the region and calculating the area, if the region is already mapped it can be more

convenient to measure the area directly from the map. If the region is an irregular shape, a

planimeter can be used to measure the area. A surveyor uses a planimeter to trace the region

on a map. The difference between the final and initial reading of the planimeter is the area of

the region traced (on the map). Once this area is measured, the map scale is used to convert

the area into the correct units for the actual region (not on the map). This method saves time

compared to physically measuring the region and calculating the area.

For some projects, soil (or other material) must be added to or removed from a site

based on elevation needs. To calculate the volume of cut (material to be removed) or fill

(material to be added), surveyors can look at areas of cross sections of a project. After

calculating the areas of cross sections at each end of a station, surveyors take the average of

the two areas and multiply that by the length of the station (or the length until volume switches

between cut and fill). These volume calculations let surveyors know where the need additional

material, where they need to remove material, and whether they will need to borrow or have

excess material.

Task 1: Measuring Area Using a Planimeter

Procedure:

Before any measurements were taken, each group member handled the planimeter and

made sure everyone knew how to read the instrument. Each planimeter has a constant

associated with it that is used to calculate areas of regions. Before taking any measurements,

the group had to determine the planimeter constant. To do this, a square of known area was

drawn (in this experiment the square was 3”x3”). The square was then traced 3 times with the

planimeter and the number of revolutions the planimeter made each time was recorded. The

mean and standard error of the mean of the number of revolutions were calculated (see

Appendix A). Using the equation (Area) = (Constant) x (Number of revolutions), the constant

could now be solved for using the known area of the square and the mean of the number of

revolutions.

With the constant of the planimeter now known, the planimeter could be used to

measure an unknown area of a region. Given a map with a region outlined, the region was

measured 3 times with the planimeter and the area was calculated each time, along with the

mean and standard error of the mean for the area (see Appendix A).

Because the planimeter was being used to measure a region on a map, the area

measurements were in units of square inches. The map scale allowed the group to convert this

area on the map (in units of square inches) to units that correspond to the actual region (for this

lab, acres). The group then converted the mean area measured with the planimeter to acres

(see Appendix A).

Results:

Planimeter Constant = 9.8901

Standard Error of the Mean of Planimeter Constant = 0.00569

Region on Map: Area Calculations

Reading 1 Reading 2 Reading 3 MeanStandard

Error of the Mean

Planimeter Reading 0.511 0.456 0.538 0.502 0.0241

Area on map (in2) 5.05384 in2 4.50989 in2 5.32087 in2 4.962 in2 0.3656 in2

Area of actual region (acres) 1.16 acres 1.04 acres 1.22 acres 1.14 acres 0.08165

acres

Table 1: The results from Task 1

Task 1 Analysis:

Knowing the constant for our planimeter, the area of a plot of land on a map was

calculated. The results of this test are found in Table 1. The area of the region in the map was

found to be 4.962 in^2 on the paper, and using the conversion factor of 1 in = 100 ft. we were

able to determine the area of the plot of land to be 1.14 acres. The standard error of the mean

was calculated for the area of the parcel on the paper as well as the area of the actual parcel in

acres. The standard error of the mean for the area on the paper in square inches was a bit high,

but it did not end up effecting the standard error of the mean for the area of the parcel in acres.

The map of the parcel of land, the planimeter readings used in the calculations, and the

calculations themselves can be found on the second page of Appendix A.

Task 2: Measuring Area Using a Planimeter

Procedure:

In this part of the Lab we were given information that described the cross sections at

three stations, 100 ft. apart, for a two lane roadway, and were asked to calculate the cumulative

earthwork quantities at each station. In order to do this, the cross sections were first drawn out

and coordinates plotted for each point on the section. These coordinates were then put in a

tabular format and the coordinate method for finding area was utilized to determine the area of

each cross section. All of this information is located on the third and fourth pages of Appendix A.

Using the area for the cross section at each station, the average end area volumes of cut and fill

were calculated for the space between stations 1 & 2 and 2 & 3. These volumes were then used

to calculate the cumulative earthwork quantities at each station - assuming station 1+00 started

with 0 yd3 of material.

Results:

Cross Section Areas:

A1+00: 85.02 ft2 (cut)

A2+00: 70.55 ft2 (fill)

A3+00: 140.50 ft2 (fill)

Average End Area Volume:

Ve 1-2 (cut): 157.447 yd3 (cut)

Ve 1-2 (fill): 156.778 yd3 (fill +20% for shrinkage)

Ve 2-3 (cut): 0 yd3 (cut)

Ve 2-3 (fill): 468.989 yd3 (fill +20% for shrinkage)

Cumulative Earthwork Quantities:

Ve 1+00: 0 yd3

Ve 2+00: 0.669 yd3 (waste material)

Ve 3+00: 468.32 yd3 (fill material)

Task 2 Analysis:

The cross sections for Task 2 can be found on the third and fourth pages of Appendix A.

Using the coordinate method, the area of each section was determined, and the results of these

calculations can be found above. The tables of coordinates that were used to calculate the area

of each section are located on the third and fourth pages of Appendix A.

Using these cross sectional areas, the average end area volumes in cut and fill were

calculated between the stations. To adjust for shrinkage, 20% was added to the volumes in fill.

These calculations can be found on the fourth page of Appendix A, and the results are listed

above in the Task 2 Results section.

The cumulative earthwork quantities were then calculated, assuming that at station 1+00

we started with 0 yd3 of material. Between stations 1+00 and 2+00 the quantities of cut and fill

needed were almost equal, so the area between stations 2+00 and 3+00 will require the vast

majority of the fill material needed for the project. Overall, 468.32 yd3 of fill material will be

needed to complete work on this project. The cumulative earthwork quantities at each station

can be found above, and the calculations are located on the fourth page of Appendix A.

Conclusions

In this lab, the planimeter was used to determine the area of a given parcel of land by

determining the planimeter constant using an area of known volume. This tool is very useful in

determining the area of a plot of land because it requires only an aerial photograph or scaled

map of the plot. In Task 2, the cumulative earthwork quantities were calculated for a two lane

roadway divided by 3 stations. Using the cross section at each station, the volume of earthwork

in cut and fill between each of the stations was calculated and used to determine the cumulative

earthwork quantities needed at each station for the entire project. These types of earthwork

calculations can be very useful in future surveying labs and projects.

Appendix A:

Appendix A cont:

Appendix A cont:

Appendix A cont: