Grab a pin, a pendulum (weight on the end of a piece of string), a ruler and pencil. Push the pin through the object and attache the pendulum to the pin. Make sure that the irregular object hangs freely on the pin. Allow pendulum and object to settle. Carefully mark where the pendulum crosses the bottom edge of your object, then join that mark with the original hole using a ruler.
If you repeat this experiment the lines should all cross at the cengre of gravity because that will always come to rest under the pin.
This assumes that the flatness is uniform.
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If you wanted to describe the length and breadth of an irregular shape you would find the maximum length and breadth. You could sandwich the object between two flat boards, and measure the distance between the boards. Then rotate the object 90o and do it again, for the breadth.
Irregular objects can have their perimeters measured in a variety of different ways. Probably the easiest is to run a string along the edge of the object and then later lay the string flat and measure it. Another way is to measure in partial straight segments and approximate.
A round object with flat, circular ends is a cylinder. It is shaped like a can of vegetables.
A regular octagon cannot tile a flat surface, it needs squares as fillers. An irregular octagon can tile a flat surface alone.
A flat round object is a circle. Surface area of a circle = pi*radius squared.