enclosing-ball slimness factor is: It is also known that: where c is any side of the triangle and A,B are the adjacent angles. Hence, for an obtuse triangle with acute angles A and B (and longest side c), the enclosing-ball slimness factor is: Note that in a right triangle, , so the two expressions coincide. Two-balls slimness The inradius r and the circumradius R are connected via a couple of formulae which provide two alternative expressions for the two-balls slimness of an acute triangle: For an obtuse triangle, c/2 should be used instead of R. By the Law of sines: Hence the slimness factor of an obtuse triangle with obtuse angle C is: Note that in a right triangle, , so the two expressions coincide. The two expressions can be combined in the following way to get a single expression for the two-balls slimness of any triangle with smaller angles A and B: To get a feeling of the rate of change in fatness, consider what this formula gives for an isosceles triangle with head angle θ when θ is small: The following graphs show the 2-balls slimness factor of a triangle: Slimness of a general triangle when one angle (a) is a constant parameter while the other angle (x) changes. Slimness of an isosceles triangle as a function of its head angle (x). Fatness of circles, ellipses and their parts The ball-based slimness of a circle is of course 1 - the smallest possible value. For a circular segment with central angle θ, the circumcircle diameter is the length of the chord and the incircle diameter is the height of the segment, so the two-balls slimness (and its approximation when θ is small) is: For a circular sector with central angle θ (when θ is small), the circumcircle diameter is the radius of the circle and the incircle diameter is the chord length, so the two-balls slimness is: For an ellipse, the slimness factors are different in different locations. For example, consider an ellipse with short axis a and long axis b. the length of a chord ranges between at the narrow side of the ellipse and at its wide side; similarly, the height of the segment ranges between at the narrow side and at its wide side. So the two-balls slimness ranges between: and: In general, when the secant starts at angle Θ the slimness factor can be