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Equivalently, a square with this distance as its side length has the same area as a rectangle with the two heights as its sides. Then, the construction of a circle with the top of the painting diametrically opposite the reflected bottom, and its intersection with the line at eye level, follows Euclid's Book II, Proposition 14, which describes ...
Solid angles can also be measured in square degrees (1 sr = (180/ π) 2 square degrees), in square arc-minutes and square arc-seconds, or in fractions of the sphere (1 sr = 1 / 4 π fractional area), also known as spat (1 sp = 4 π sr). In spherical coordinates there is a formula for the differential,
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 ...
Squaring an integer, or taking the area of a square with integer sides, results in a square number; these are figurate numbers representing the numbers of points that can be arranged into a square grid. [7] Since four squared equals sixteen, a four by four square has an area equal to its perimeter. That is, it is an equable shape. The only ...
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 ...
Among all quadrilaterals with a given perimeter, the one with the largest area is the square. This is called the isoperimetric theorem for quadrilaterals. It is a direct consequence of the area inequality [38]: p.114 where K is the area of a convex quadrilateral with perimeter L.
The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with respect to an arbitrary axis. The unit of dimension of the second moment of area is length to fourth power, L 4, and should not be confused with the mass moment of inertia.
Quadratrix as a plane curve for side length =, as given by the parametric formula for −∞ < t < ∞, with singularities when t is a nonzero integer multiple of π. If one places such a square with side length in a (Cartesian) coordinate system with the side ¯ on the -axis and with vertex at the origin, then the quadratrix is described by a parametric equation that gives the coordinates of ...