To find the area, first divide the shape into regular, simple shapes. Then use formulas to find the area of the smaller, regular shapes. Lastly, add up all the smaller areas to find the area of the original shape.
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A container of liquid in which to immerse the shape.
You would use a tool to measure the lengths of the sides or radii of the area and then use a formula to convert the lengths into an area. The tool would depend on the size of the object and the precision required. The tool could be an electron microscope, or a ruler, or a telescope and in general there would be no difference between a tool used to measure in the metric system or the imperial system. As an example of an unusual tool to measure area... in certain types of chemistry, the chemist needs to measure the area under a peak on a graph, and the traditional way of doing that is to cut the peaks out of the graph and weigh it on an analytical balance. Assuming the paper is fairly consistent in terms of density, the ratio of the weights is the same as the ratio of the areas.
If you want to measure it's volume, place it in a measuring cylinder containing a certain amount of water, and measure the extra units to get the volume of the object
A graduated cylinder.
A calliper is any tool with a claw-like shape/movement, like a calliper micrometer, which is used to measure the size of small parts with high accuracy, or the calliper brakes on a bicycle or a car with disc brakes.