Distance and Area
Distance and area are fundamental concepts in measurement that quantify space and dimensions. Distance refers to the absolute space between two points, expressed in various units such as meters or feet, with the International System of Units (SI) standardizing these measurements for accuracy and precision. Area, on the other hand, is a derived unit that measures the extent of a two-dimensional surface, calculated using distances; for example, the area of a rectangle is determined by multiplying its length and width, resulting in square meters (m²).
Understanding distance is essential in contexts like travel, where accurate measurements can differentiate between direct paths and longer routes. Area calculations are vital in practical applications, such as real estate, where square footage informs buying and selling decisions, or in home improvement projects that require surface area estimates for painting or flooring. Different shapes necessitate distinct formulas for area, highlighting the importance of geometry in spatial analysis. In summary, these concepts play a crucial role in both everyday life and various professional fields, enabling effective communication and informed decision-making regarding physical spaces.
Distance and Area
FIELDS OF STUDY: Classic Mechanics
ABSTRACT: Distance and area are some of the most useful measurements and calculations to make in everyday life—whether calculating the square footage of a room or measuring a desk. Starting with one-dimensional measurements of distances, one can calculate the area of various two-dimensional shapes and the total surface area of three-dimensional objects. Though the formulas used to make these calculations were devised centuries ago, they are still in wide use.
PRINCIPAL TERMS
- accuracy: how closely a measurement of a property matches the real value.
- derived unit: unit of measure that is described in terms of two or more base units; for example, meters are a base unit of distance, whereas square meters are derived units of area calculated from meters of height multiplied by meters of depth.
- displacement: the absolute distance between two points.
- International System of Units (SI): also known as the "metric system"; a widely used standardized system of units for measuring natural phenomena.
- measurement: quantifying an observation (e.g., the length of a person’s foot) using discrete units (e.g. meters); alternately, the unit or system used to do so.
- meter: the SI base unit of distance (or length) measurement.
- precision: how well a measurement agrees with other measurements of the same phenomenon.
- scale: a description of the area under observation in broad approximation.
Measuring Spaces
The distance between two points could be the distance between two cities, the distance between two ends of a ruler, or the distance between the bottom of one’s foot and the top of one’s head. In these cases, "distance" is equivalent to length, width, and height. It is always a quantitative measurement of the space between two points. In the International System of Units (SI), the meter (m) is the standard unit in which distances are measured. Americans commonly use feet (ft) instead of meters for making measurements. One meter is equal to approximately 3.28 feet.
The foot was widely used in many countries before the advent of the metric system. But it varied in its value by location. This created a problem. In their local areas where everyone was using the same standard, measurements may have resulted in high precision. However, their accuracy would be questionable—especially when translating measurements between locations that had different ideas of what a foot meant. The standardization of the SI has helped to ensure that scientists and engineers are able to make measurements of maximum accuracy and precision without worrying that the units they are using are a source of uncertainty. At larger scales of observation, units of kilometers (km) or miles (mi) may be used instead of meters or feet. At smaller scales, centimeters (cm) or inches (in) may be used.
Sometimes, it is worth making distinctions between the actual distance traveled and the distance between two points. Consider the winding road a car may take to get from home to the store. In this case, displacement describes the absolute distance between the car’s starting point and its end point.
Area is calculated using measurements of distance. This results in a derived unit, such as the square meter (m2). When dealing with three-dimensional shapes with multiple sides, the total area of each side is the surface area. This is also measured in square meters and calculable using well-known formulas.
Calculating Surface Area
It is possible to estimate the surface area of everyday objects by choosing a similar three-dimensional geometric shape and applying the mathematical formulas for the surface area of that shape. For instance, a pencil can be considered a cylinder.
The process of determining the surface area of a shape begins with taking measurements of its length, width, and height. After taking these measurements, the values are plugged into the formula for the appropriate shape to get an approximation of the object’s surface area.
For more complicated shapes, two options exist: if a shape can be broken down into component shapes, the surface area can be estimated. Consider a spire. It is, roughly speaking, equivalent to a cylinder with a cone on top. The surface area could be estimated by calculating the surface area of the cylindrical and conical portions and then subtracting the value of the connecting circular surface from each.
Estimating the Surface Area of a Tire
Given a tire of a 0.75-meter radius (r) and a 0.2-meter tread width (w), derive the circumference in meters. This will allow for an estimate of the surface area in square meters.
To figure out the circumference, one must know the formula for the circumference of a circle. Circumference (C) equals two multiplied by pi (π) multiplied by the radius.
C = 2πr
C = 2π (0.75 m)
C ≈ 4.712 m
To estimate surface area, one must first decide on a three-dimensional geometric shape that best approximates a tire. In this case, a short cylinder works best (imagine the tire is laid on one of its flat sides). The formula for the surface area (A) of a cylinder is the circumference (C, or 2πr) of one of the circular ends, multiplied by the height (h) of the cylinder, plus two multiplied by the area of each of the circular ends (2πr2):
A = 2πrh + 2πr2
In this case, the value of h is the same as the width of the tire. Thus, h is equal to 0.2 meters, giving:
A = 2π (0.75 m)(0.2 m) + 2π (0.75 m)2
A = 2π (1.5 m2) + 2π (0.5625 m2)
A ≈ 4.48 m2
Area in the Everyday
The ability to estimate area and surface area is useful in a variety of real-world scenarios. For example, in real estate, understanding area helps one make sense of square footage, which tells how much living space there is. If one is selling, buying, or renting a house or apartment, one can calculate the square footage. If one wants to paint a room, estimating the surface area of the walls will allow one to determine how much paint to purchase.
Bibliography
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Klein, Herbert Arthur. Science of Measurement: A Historical Survey. 1974. New York: Dover, 2012. Digital file.
Somervill, Barbara A. Distance, Area, and Volume. Chicago: Heinemann, 2011. Digital file.
Thompson, A., and B. N. Taylor. "The NIST Guide for the Use of the International System of Units." National Institute of Standards and Technology. US Dept. of Commerce, 5 Oct. 2010. Web. 25 Mar. 2015.
Wagner, Mark. The Geometries of Visual Space. Mahwah: Erlbaum, 2006. Print.
Weinstein, Eric W. "Surface Area." MathWorld. Wolfram Research, 19 Mar. 2015. Web. 25 Mar. 2015.