# Body volume and surface area investigation report

Introduction

In this investigation the aim was to get an estimate of the volume and surface area (and a real volume and surface area ) of body parts represented by various shapes including: sphere, cylinder and rectangular prism.The way in which you calculate the volume of a cylinder is to times pi by the radius squared times by the height.(π x r3 x h.) A sphere's volume can be calculated by dividing 4 and 3 times by pi times by radius cubed.(3 ). And a rectangular prism's volume can be calculated by timing length times by width times by height.(). The conjecture is that the real and estimated volume and surface area in this investigation will be different because the estimate is just estimating the volume and surface where as the real volume and surface area is actually calculating the real volume and surface. Mathematics being utilised within this investigation include: volume , surface area, pi, multiplication, addition, mass, height, radius, squaring and cubing. The mathematics being utilised in this investigation would be applied in real world circumstances such as: finding the area of a piece of land (architect and anything to do with building like a city planner etc.) ,finding the volume of a tank of some sort such as a tank at a wine distillery and planning crop fields by calculating how much area is needed.

Mathematical calculations -

Real total volume calculations

Formula =

Body mass (kg) = 75 kg

75 x 1000 = 75,000 cm3

Real total volume calculations

Formula =

Body mass (kg) = 75 kg

Height (cm) = 170

75 x 170 = 12750

12750 ÷ 3600 = 3.54

3.54 x 10000 = 35416.6 cm2

Sphere volume calculations (Head)

Formula = 3

r3 = 843.9

4/3 = 1.3

1.3 x 3.14 = 4.08

4.08 x 843.9 = 3443.11 cm3

Cylinder volume calculations (Neck)

Formula = 3.14 x r2x h

r2= 79.21

3.14 x 79.21 = 248.71

248.71 x 15 = 2939.5 cm3

Rectangular prism volume calculations (Torso)

Formula =

Length = 31.4

Width = 19.2

Height = 47.6

31.4 x 19.2 = 602.8

602.8 x 47.6 = 28693.2cm3

Sphere surface area calculations (Head)

Formula = 2

r2= 89.3

4 x 3.14 = 12.56

12.56 x 89.3 = 1121.6 cm2

Cylinder surface area calculations (Neck)

Formula = 2

r = 8.9

r2 = 79.21

h= 15

2 x 3.14 = 6.28

6.28 x 8.9 = 55.8

55.8 x 15 = 838.38

2 x 3.14 = 6.28

6.28 x 79.21 = 497.4

838.3 + = 1335.7 cm2

Rectangular prism surface area calculations (Torso)

Formula =

Length = 31.4

Width = 19.2

Height = 47.6

19.2 x 31.4 = 602.8

602.8 + 47.6 x 31.4 = 2097.4

2097.4 + 47.6 x 31.4 = 3011.3

3011.3 x 2 = 6022.7 cm2

## Analysis

The real and estimated calculations are similar in some ways but different in other ways. Some of the estimated volumes were underestimated because they were lower than what the real volume was. For example, the volume for the torso (rectangular prism) was estimated to be 27683 cm3 but the actual volume turned out to be 28693.2 cm3 therefore the estimated volume was in fact underestimated.The estimated surface area stays pretty consistent with the real surface area with the exception of torso. The estimated surface area for the torso was at 5878 cm2 with the real surface at 6022.7 cm2,therefore the estimated surface area was underestimated. The shapes utilised within this investigation definitely could have altered what the volumes and surface areas are because of their weak representation to irregular shapes found in the human body. If more irregular and realistic shapes were used in this investigation there would been more accurate results due to the fact our bodies aren’t actually made up of shapes used in the investigation, therefore making the formulae and calculations utilised within this investigation void. The conjecture was supported because there were many differences between the estimated volumes and surface areas.

## Discussion

In this investigation it was assumed that the shapes utilised accurately represented the shapes within the human body. With these assumptions there was an assumption that the volume and surface area of the body could be calculated with the same formula as the shapes that represented the various body parts. By using shapes that are not used within our bodies caused there to be limitations in how accurate the results would be. The limitations are plentiful within this investigation such as: limitations in accuracy due to unrealistic shapes being used, inaccurate measuring due to human error could have possibly caused the results to be more inaccurate and the lack of knowledge in calculating volume and surface area in some shapes might of caused errors within the calculations. The methods for minimising the impact of these assumptions and limitations are: using a more accurate measuring tool than a piece of string and a ruler, utilising more realistic shapes to get a more of an accurate result than using a cylinder, spheres etc. and gaining knowledge of his to calculate the volume and surface area of certain shapes to further ensure the results will be as accurate as can be.

## Conclusion

In conclusion the results were consistent with some of the estimated volumes and surface areas being underestimated such as the torso (rectangular prism).The conjecture was supported because the estimated and real volume and surface turned out to be quite different due to the fact the estimates were only an estimate of the real thing, although some of the estimates were close to the real results. The aim was achieved as the total estimated and real volume and surface area were found. The total real and estimated volume and surface area were calculated and the working out for those calculations have been included underneath the table as well as example calculations of all the types of shapes used within the investigation.