Calculations for what is called here a spindle. Such a spindle is created when a circular segment rotates around its chord. It is therefore a solid of revolution. This solid has two vertices. The straight cross-section perpendicular to the axis of rotation of the spindle is always a circle. The largest circle is the cross-section through the center. A cross-section along the axis of rotation is a shape referred to here as a pointed oval.
Enter the radius of the circle of the original circular segment and the height of this segment. Choose the number of decimal places, then click Calculate. a is the distance of the chord of the circular segment from the parallel diameter line of the circle.
Formulas:
pi:
Radius, height and distance have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit -1.
A lemon has simplified roughly this shape. A spindle is a rotating needle on a loom from which the weaving thread is unwound. It has approximately this geometric shape when more thread is wound in the middle of the spindle than at the ends. The word spin means to rotate. Therefore, the name spindle seems appropriate for this geometric shape, even though it is not a mathematically established term. The term also appears at a spindle torus, which is a circle that rotates about an axis within itself.
Another solid of revolution of the circular segment is the spherical cap, which is much more common and easier to calculate. In this case, the circular segment does not rotate around its chord, but around the perpendicular bisector to it. A spherical cap can be easily cut out of a sphere, a spindle cannot.
The derivation of the formulas for surface area and volume is mathematically demanding. They are obtained through integration, the surface area from the rotation of a circular curve and the volume from integration over areas. The integral formulas are: