Introduction

Clay tablets from Mesopotamia and papyri from Egypt provide evidence that work with area has been part of mathematics since its early history. These Ancients knew how to find areas of squares, circles, triangles, trapezoids, and a number of other shapes for which we no longer have names.

Like many other physical quantities, we usually measure area indirectly. That is, we measure something else, such as lengths, a radius, or angles. Then we do some calculations to find area based on appropriate formulas. The object determines the formula we use and the measurements we make.

There are a number of methods to measure volumes. Some of them are indirect, as for areas, involving linear and angular measurements and the use of formulas. Interestingly, some of them are more direct. One example is the measurement of liquid when cooking, using calibrated measuring cups. Similarly, the volume of a non-porous solid can be measured by submerging it in water to see how much water it displaces (it helps if the solid doesn't float).

Many instruments are cleverly designed devices that convert a value we want to measure into some scale that we can read directly. For example, thermometers (the bulb type, not the digital ones) convert the volume of the liquid (this used to be mercury, but now it is usually tinted alcohol) in the bulb into a length. When the liquid expands or contracts with the temperature, the calibrations along the tube let us read the temperature that corresponds to volume.