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The radius of the first arc must be chosen large enough to cause all successive arcs to end on the correct side of the next crossing point; however, all sufficiently-large radii work. For two lines, this forms a circle; for three lines on the sides of an equilateral triangle, with the minimum possible radius, it forms a Reuleaux triangle, and ...
In the following equations, denotes the sagitta (the depth or height of the arc), equals the radius of the circle, and the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides of a right triangle with r {\displaystyle r} as the hypotenuse , the Pythagorean ...
Draw a circle of radius OA and center W. It intersects the original circle at two of the vertices of the pentagon. Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of the pentagon. The fifth vertex is the rightmost intersection of the horizontal line with the original circle.
Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or ...
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 nine-point circle is tangent to the incircle and excircles. In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: [28] [29] The midpoint of each side of the triangle; The foot ...
where A, B, C denote both the triangle's vertices and the angle measures at those vertices; H is the orthocenter (the intersection of the triangle's altitudes); D, E, F are the feet of the altitudes from vertices A, B, C respectively; R is the triangle's circumradius (the radius of its circumscribed circle); and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C ...
The radius of the inscribed circle of an isosceles triangle with side length , base , and height is: [16] 2 a b − b 2 4 h . {\displaystyle {\frac {2ab-b^{2}}{4h}}.} The center of the circle lies on the symmetry axis of the triangle, this distance above the base.
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