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Application of Pythagoras:Shaded Region

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Explanation:

A circle is a simple shape of Euclidean geometry consisting of the set of points in a plane that is a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.

Circles are simple closed curves which divide the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.

A circle is a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.

 

Area enclosed

Area of the circle = π x area of the shaded square

As proved by Archimedes, the area enclosed by a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to π multiplied by the radius squared:

(Our solved example in mathguru.com uses this concept).

 

Equivalently, denoting diameter by d,

that is, approximately 79 percent of the circumscribing square (whose side is of length d).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

 

http://en.wikipedia.org/wiki/Circle

 

Rectangle

In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as Rectanglen.PNG ABCD.

A so-called crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. Its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.

If a rectangle has length l and width w

It has area A = lw (Our solved example in mathguru.com uses this concept).

 

http://en.wikipedia.org/wiki/Rectangle

 

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.