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It depends on what the shape is. There are relatively simple formulae for triangles, circles, parallelograms, regular polygons; and more complicated formulae for ellipses, irregular polygons and other shapes.

If the shape can be described algebraically, then integration may provide an answer.

Then there are non-analytical methods for very irregular shapes. You can get an estimate by mapping the shape onto a grid and counting the number of squares covered and part covered. Or you can cut out an identical shape from a uniform lamina and compare its mass to the mass per unit area.

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