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Areas of Plane Figures

• The area A of a square is the square of the length $\ell$ of any one of its 4 sides: $A = \ell\,^2$. In fact the question of which word "square" is named after which is a sort of chicken vs. egg problem for which there is no logical resolution (even though there may be an historically correct etymological answer).

• The area A of a rectangle (a bit more general) is the product of the length b of a long side ("base") and the length h of a short side ("height"): A = bh.

• The area A of a triangle with base b and height h (measured from the opposite vertex down perpendicular to the base) is $A = {1 \over 2} bh$. (This is easy to see for a right triangle, which is obviously half a rectangle, sliced down the diagonal. You may want to convice yourself that it is also true for "any old triangle.")

• The area A of a circle of radius r is given by $A = \pi r^2$ where $\pi$ is a number, approximately 3.14159 [it takes an infinite number of decimal digits to get it exactly; this is because $\pi$ is an irrational number^4.3 -- i.e. one which cannot be expressed as a ratio of integers], defined in turn to be the ratio of the circumference $\ell$ of a circle to its diameter d: $\pi = \ell/d$ or $\ell = \pi d$.

Were you able to visualize all these simple plane (2-dimensional) shapes "in your head" without resort to actual drawings? If so, you may have a "knack" for geometry, if not Geometry. If it was confusing without the pictures, they are provided in Fig. 4.1 with the appropriate labels.

  

Figure: A few plane geometrical shapes, with labels.