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Another proof that uses triangles considers the area enclosed by a circle to be made up of an infinite number of triangles (i.e. the triangles each have an angle of dπ at the centre of the circle), each with an area of β 1 / 2 β · r 2 · dπ (derived from the expression for the area of a triangle: β 1 / 2 β · a · b · sinπ ...
The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane.
The same formula holds for any three-dimensional objects, except that each should be the volume of , rather than its area. It also holds for any subset of R d , {\displaystyle \mathbb {R} ^{d},} for any dimension d , {\displaystyle d,} with the areas replaced by the d {\displaystyle d} -dimensional measures of the parts.
where a is the radius of the circle, (,) are the polar coordinates of a generic point on the circle, and (,) are the polar coordinates of the centre of the circle (i.e., r 0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x axis to the line connecting the origin to the centre of ...
The centre of a circle is the point equidistant from the points on the edge. Similarly the centre of a sphere is the point equidistant from the points on the surface, and the centre of a line segment is the midpoint of the two ends.
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius .
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Gauss's circle problem asks how many points there are inside this circle of the form (,) where and are both integers. Since the equation of this circle is given in Cartesian coordinates by x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} , the question is equivalently asking how many pairs of integers m and n there are such that