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When I saw the derivation of area of square, it was by dividing the square into two triangles and then adding them. When I saw the derivation for triangle, it was just the opposite, dividing a square into two traingles. Which is the first shape we can derive the formula of from which the formula for other shapes can be obtained? Thank You.

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I think your question is quite vague. What do you mean by "the first shape"? I mean technically you could take any shape and divide it up to make other shapes.

The best thing to do is to take what is appropriate for the task at hand from what you already know. The reason triangles, and squares are used is because they're quite simple shapes to calculate.

For example if you can see a shape is composed of circles and rectangles, you'd be mad to try and work out the area of the circular area by adding more triangles over and over if you already know the formula for the area of a circle as you'd be over-complicating the task at hand.

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The most intuitive way to establish the formulas for areas is by considering square grids and counting the cells in rectangles of various sizes. This way you easily observe that the area is the product of the width by the height.

You can also work this out by filling a triangle with an arrangement of triangles, but this is a much less natural and straightforward process.

IMO, the rectangle (rather than square) comes first.

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Square (or rectangle) comes first with area got by squaring from (multiplying) the sides and a triangle is perceived having its area half only.. later in comparision.

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