Home Contact Sitemap

Chapter 8. Solid geometry

Useful Calculations

Useful Calculations

The purpose of this work is to promote fluency in the language of mathematics among students who have had little experience (or perhaps an unpleasant experience) with standard math courses. Every educated person needs to become "numerate", not only to solve practical problems, but also to develop paths of communication with scientists, physicians, and engineers, whose work requires that they think in a quantitative manner.

Volume of Cubes and Spheres

Fig. 8-1

photo credit: Wikimedia Commons

The body of a woman with a weight on her head resembles a column or cylinder, as the architect of the Erechtheion must have recognized when he supported the top of the temple (the entablature) with caryatids (Fig. 8-1). Unlike these statuesque girls, a real live woman consumes food and oxygen, and loses heat to the environment at a measurable rate; she has a metabolism. One variable that correlates well with metabolic rate is body surface area. Let us consider the surface areas of several solid figures, and compare them to human dimensions and body surface area.

The volume of a cube is simply the length of one edge, cubed : Vc = L3.If the edge is 37 cm, the volume is 373= 50 653 cubic centimeters (cc). Since the liter is nearly the same as 1 000 cc, and the average density of sea water is 1.025 kg/L, this volume of sea water weighs 51.919 kg .

The area of one face of a  cube is: L2 = 372 = 1 369 cm2. Since there are six faces, the surface of the whole cube is: 6 X 372 = 8 214 cm2

Next, consider the surfface area of a sphere of the same  volume. The volume  of a sphere is:

Vs = π (4 /3) r3

(4/3 = 1 1/3 = 1.333...)

r3 = V/1.333π = 50 653 cc/ 4.188790205 = 12 092.513 cc.

r =  3√12092.513 = 22.953 cm .

This is the radius of a sphere with the same volume as the cube above (Fig. 8-2). The surface of this sphere is:

As = 4π r2 = 4 x (22.953 cm)2  =  6 620.469 cm2.

The ratio of surface areas, sphere / cube = 6 620.469 / 8214 = 0.8060.

This means that the sphere has 19.4% less surface than the equal-volume cube. In fact, the sphere has the least surface of any solid figure of the same volume. This is good for a storage tank, but bad for a person who wishes to lose weight, that is, become less spherical. The more spherical the body, the less surface per kg. The result is a lower weight-specific metabolic rate, and therefore a slower rate of weight loss.

Exercise 8-1.

(a) A small bedroom is nearly cubical. The floor is 3.0 m each side, and the ceiling height is 2.60 m. Calculate the volume of the room.  (b) A spherical storage tank has a diameter of 3.002 m; calculate the volume.

Volume of Cylinders and People

The volume of a cylinder is the area of the base, which is circular, times the height. To construct a cylinder with the same volume as the cube and sphere above, begin by selecting a base with a radius of  0.100 m.

V = π r2 h  = 50 653 cc =  0.050653 m3  = 0.031415926 h,

h  = 0.050653 / 0.031415926 = 1.612m.

The volume here was shown to weigh 51.919 kg, if the material has the same density as sea water, which the human body does.  The calculated height of the woman-shaped column is 1.612m.  The radius of this cylinder results in a circumference of  0.200 π, which corresponds to a waistline of 62.8 cm (24 ¾ in.). Her BMI = 51.919 kg/ 2.5996 m2 = 19.97 kg/m2, at the lower end of the healthy range.

The surface area of a cylinder is the circumference of the base times the height, plus the areas of the bottom and top circles. The woman-shaped column has suface area:

A = 2 π r h + 2 π r2 = 1.01306  + 0.062831853 = 1.075891853 m2

Many formulas have been invented for relating weight and height of human subjects to body surface area (BSA). One proposed by DuBois in 1916 became popular for calculating the proper dose of a drug:

S = 0.007184 x W0.425x  H0.725      

This equation uses H in cm, W in kg, and yields S in m2. For the woman-shaped column, W = 51.919 kg and H = 161.2 cm. The result of calculation is:  S = 1.53458 m2. The ratios of volume to surface (V/S) in Table 8-1 indicate that the shape of this lady does not match any of these three figures of solid geometry, but approaches the cylindrical.

Table 8-1. Figures with the same volume (0.050653 m3 = 50.653 L).
area  (m2) V/S (L/m2)

sphere

0.662045

76.51

cube

0.821400

61.67

cylinder

1.076310

47.06

human body

1.534580

33.02

Use of the equation above produces the result that a 4.536 kg (10 lbs) increase in body weight, with no change in height, increases weight by 8.73% and surface by 3.62 %. The body has become more spherical;metabolic rate, since it is proportional to surface area, would also increase less than weight.

The classic BSA equation, still in use, over-estimates the surface of obese people. A recent revision changes the exponents, and gives the result in cm2:

S = 94.9 X W0.441 X H0.655

Exercise 8-2.

With the revised equation, re-calculate the BSA of the woman-shaped column. To get further practice with the yx function on the calculator, find BSA by both equations for John Godina, who is 193.0 cm tall, and weighs 129.0 kg.

Like a fossil in the rocks,  the cubic foot remains stuck in the literature. To convert:

1 m = 39.37 inches, 1 m3 = (39.37)3 = 61 023.37795 in3

1 ft = 12 in, 1ft 3 = (12)3 =  1 728 in3

61 023/1728  = 35.314 cubic feet = 1 cubic meter

Exercise 8-3.

(1) Convert the volumes in Exercise 8-1 to cubic feet.

(2)  If a cylinder has the same radius as a sphere, and a height of 2r, then the sphere fits inside the cylinder like a tennis ball in a can that holds only one ball. Use the equations above to find what fraction of the can volume is occupied by the ball.

The last problem was solved by Archimedes, who asked that the sphere-in-cylinder figure be inscribed on his tombstone. Look for it, when you visit Syracuse.