The area of irregular shapes of known density and thickness can be estimated by measuring their mass with an analytical balance.

Measuring area seems to be a pretty straightforward task, something we probably all did our fair share of in primary school.  But what do you do when the shape is not a simple ellipse or polygon.  Philip Gabriel of Colorado State University has published a method to introduce high school students to the idea of inferring area by measuring the mass of an object with known density and thickness.  The method has the students estimating area for a variety of arbitrary shapes.  To get an accurate estimate of area, of course they measure the mass with an analytical balance.