Estimating Area

TASK. In medicine, calculation of body surface area is sometimes very important. For example, severe burns are usually described as covering a percentage of the body surface area. Some chemotherapy drug dosages are based on body surface area. How might body surface area be measured? What factors influence the accuracy of the estimates?

COMMENTARY. Three main mathematical themes could be emphasized in this task: estimation, partitioning, and successive approximation. These need to be combined with an understanding of the relationships among measurements of distance, area, volume, and weight. This task might also be used as an introduction to the calculus topic of integration.

In everyday life, we estimate area in order to determine how much paint to use in painting our homes, how much carpet to buy for a room, how many plants to buy for a garden, or how much grass seed or fertilizer to buy for our lawns. Some approaches to this task can also lead to discussion of proportion and scale. Highway designers, landscape designers, interior designers, and architects all make and interpret diagrams drawn to scale.

An important component of this task is estimating the error in measurement--perhaps finding both upper and lower bounds, which lays some of the groundwork for calculus. In some cases, upper and lower bounds may lend themselves to further refinement.

MATHEMATICAL ANALYSIS. There are many possible avenues of approach for this task. One possibility is to consider the human body as a collection of cylinders. Each limb, the head, and the midsection, for example, might be approximated as cylinders. Finding the sum of the lateral surface area of each of these would give a good first approximation. This procedure could be refined by adding the areas of the "ends" of the cylinders and subtracting the areas where the cylinders are attached together. The procedure could be further refined by considering, for example, the head and the neck to be a sphere and a cylinder, respectively, and eventually by approximating the fingers as individual cylinders.

A very different approach would be to take pictures of a person--front, lateral, and top views--and to superimpose a flat grid on the pictures to approximate the surface area. This approach would require additional discussion of proportions due to the scale of the picture. The accuracy would depend not only on the size of the grid relative to the dimensions of the person in the picture, but also on the reasonableness of projecting a three-dimensional human onto two-dimensional pictures to estimate surface area. Using smaller and smaller grids to achieve successively better approximations foreshadows some of the ideas of calculus.

An approach that also foreshadows calculus but doesn't involve scaling is to cover the body with patches of cloth of known area. If the pieces of cloth are all the same size, the accuracy of the approximation would depend on the size of the pieces. Covering the body with cloth suggests a very elegant approach that doesn't involve calculus at all. If the entire body is clothed with close-fitting, non-stretch cloth of consistent thickness and density, the surface area may be determined by weighing the cloth and then dividing by the weight of a piece of cloth of unit area.

From any of the above approximations, students would be afforded experience and data from which to discuss the adequacy of the standard medical practice of approximating surface area by using the following formula:

where height is measured in centimeters and weight is measured in kilograms. The result of the calculation gives an approximation for body surface area in square meters. Thus, for example, a person who is 5'10" (177.8 cm) tall and weighs 180 lbs. (81.8 kg) has a surface area of approximately 177.8 x 81.8/3600 x 2 square meters. This calculation is based in part on the assumption that, for humans, weight is roughly proportional to volume, an assumption that is also worthy of investigation. The formula has a certain dimensional consistency in that the product of height (one-dimensional) and weight (three-dimensional) gives a four dimensional quantity. Taking the square root then gives a two-dimensional result, consistent with the fact that surface area is a two-dimensional quantity.

EXTENSIONS. Given a map of the United States, students might estimate the area in square miles of the states of Colorado, Texas, Florida, and Vermont and compare their estimates to the actual area of each state given in an almanac or an encyclopedia. The students might discuss the accuracy of their estimates and why the process is easier for some states than for others. If the students are using a square grid to do the approximation, they can ask: What size grid is needed to estimate the area to a given accuracy?

Using a map causes another kind of measurement error. Because a map is a flat representation of the curved surface of the earth, there is some inevitable error in the way it shows angles or areas. How big an area on the earth must be considered before such errors are measurable? Understanding the reason for this distortion and some of the projections used by map makers involves solid geometry and spherical trigonometry.

Students also could explore the extent to which a state's topography affects its surface area. Is the surface area of a 100-mile square section of western Colorado the same as a 100-mile square section of eastern Colorado? This leads to interesting questions about how surveyors actually calculate the area of steeply sloped land.

Other extensions might concern volume or perimeter, estimating, for instance, the amount of air in a school building to determine how often the ventilation system refreshes the air. Approximating the perimeter of a territory can lead to some interesting findings. When the coastline of Britain on a geographical map is approximated by line segments corresponding to 500 km on the map, the result is 2600 km. When the coastline is approximated by segments corresponding to 17 km, the result is 8640 km--more than a three-fold increase. In contrast, when the border of Utah is measured in the same ways, the estimated length goes from 1450 km to about 1890 km--not even a two-fold increase. Fractal dimension is concerned with characterizing differences like these (Peitgen et al., 1992).

References

Peitgen, H.-O. et al. (1992).

Fractals for the classroom. New York: Springer-Verlag.