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Following Archimedes' argument in The Measurement of a Circle (c. 260 BCE), compare the area enclosed by a circle to a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius. If the area of the circle is not equal to that of the triangle, then it must be either greater or less.
The area within a circle is equal to the radius multiplied by half the circumference, or A = r x C /2 = r x r x π.. Liu Hui argued: "Multiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, to yield the area of a dodecagon; if we cut a hexagon into a dodecagon, multiply its side by its radius, then again multiply by six, we get the area of a 24-gon; the ...
The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle. Given the length y of a chord and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines: = +.
The area of the circle equals π times the shaded area. The area of the unit circle is π. [153] π appears in formulae for areas and volumes of geometrical shapes based on circles, such as ellipses, spheres, cones, and tori. Below are some of the more common formulae that involve π. [154] The circumference of a circle with radius r is 2πr. [155]
The arc length, from the familiar geometry of a circle, is s = θ R {\displaystyle s={\theta }R} The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of θ {\displaystyle \theta } ):
The area-equivalent radius of a 2D object is the radius of a circle with the same area as the object Cross sectional area of a trapezoidal open channel, red highlights the wetted perimeter, where water is in contact with the channel. The hydraulic diameter is the equivalent circular configuration with the same circumference as the wetted perimeter.
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
The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral. [ 36 ] The ratio of the area of the incircle to the area of an equilateral triangle, π 3 3 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}} , is larger than that of ...