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Consider completing the square for the equation + =. Since x 2 represents the area of a square with side of length x, and bx represents the area of a rectangle with sides b and x, the process of completing the square can be viewed as visual manipulation of rectangles.
The area of a rectangle is equal to the product of two adjacent sides. The area of a square is equal to the product of two of its sides (follows from 3). Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square. [10]
The area can also be calculated using the diagonal d according to =. In terms of the circumradius R, the area of a square is =; since the area of the circle is , the square fills / of its circumscribed circle. In terms of the inradius r, the area of the square is =;
That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula: [1] [2] A = s 2 (square). The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a ...
The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry. It depicts two arrangements made of similar shapes in slightly different configurations.
The fact that a rectangle of aspect ratio ρ 2 can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ 2 related to the Routh–Hurwitz theorem: all of its conjugates have positive real part.
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The area of the surface of a sphere is equal to four times the area of the circle formed by a great circle of this sphere. The area of a segment of a parabola determined by a straight line cutting it is 4/3 the area of a triangle inscribed in this segment. For the proofs of these results, Archimedes used the method of exhaustion attributed to ...
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