Session 9, Part B:
Surface Area and Volume

In This Part: Determining the Relationship | Human Measurements

 Your body's surface area is a measurement of the skin that covers your body. You may have noticed that adults and children (and babies in particular) have very different reactions to heat and cold. This happens because the body cools down by sweating at a rate proportional to the area of its skin, but warms up in proportion to its mass (volume). The ratio of surface area to mass is much larger for babies, so they cool down faster than adults. As a result, babies can catch a chill even when adults feel warm. Problem B6 Use multilink cubes to build a model of a baby and of an adult. You can use a simple, trimmed-down model, or you can build a more realistic one. Once you've completed your models, calculate the surface area-to-volume ratio for the baby and the adult.

 Problem B7 In the summer, we're warned not to leave babies or pets in cars. Yet on a hot day, an adult can sit in a car for a short period of time without harm. Use your models and mathematics to explain what is occurring in these situations, and why babies dehydrate so much more quickly than adults.

Problem B8

In Session 6, you learned that about 100 handprints will cover the body. You used your estimate of the area of your handprint to approximate your surface area. Another approach to estimating surface area is to see how much of you fits into a square.

The picture below is based on a famous drawing by Leonardo da Vinci. As shown in the picture, the person more or less fits in the square.

A person's height is approximately equal to his or her arm span (from fingertip to fingertip).

 a. Measure your height and arm span in centimeters and find the area of "your square." b. Many have suggested that three-fifths of the square is a reasonable approximation for surface area. How does three-fifths of the area of your square compare with your first approximation of your surface area based on hand size?

Problem B9

Another way to approximate a person's surface area is to use a simple formula:

Height (cm) • Thigh Circumference (cm) • 2 = Body Surface Area (cm2)

 a. Find your surface area using this formula. b. What do you think the above formula is based on? c. Do you think the above formula would work for determining the surface area of a baby? Why or why not?

 Since surface area is also related to weight, health care workers usually use a chart called a nomograph to estimate a person's surface area. To use the nomograph, a person's height (in centimeters) is located in the left-hand column, and a person's weight (in kilograms) is located in the right-hand column. These points are connected with a straight line. The surface area of a person's body is shown where the line crosses the middle scale. Note 2

 Problem B10 Use the nomograph to estimate your own body's surface area. Note that 1 kg = 2.2 lb.

 Problem B11 The density of the human body is a little greater than the density of water because of our bones and organs. One kilogram of body mass occupies a volume of about 0.9 L. Determine the volume of your body by multiplying your weight in kilograms by 0.9.

Problem B12

 a. What is your surface area-to-volume ratio? b. What is the surface area-to-volume ratio of a child who weighs 55 lb. and is 40 in. tall? c. Compare the two ratios. How do these measurements compare with your first estimates?

 This problem will be easier to solve if you convert to metric measures.   Close Tip This problem will be easier to solve if you convert to metric measures.

 "Human Measurements" adapted from Romberg, T., et al. Made to Measure. Mathematics in Context. © 1998 by Encyclopedia Britannica Educational Corporation. Used with permission. All rights reserved.
 Session 9: Index | Notes | Solutions | Video