Two-Dimensional Head Shape Calculator

Calculations for a two-dimensional head shape. This shape consists of a circular segment, comprising more than half of the circle, and a matching rectangle attached to it. The circular segment can be viewed as the head itself, and the rectangle as the neck. While head is not a standard geometric term, it seems appropriate for this shape. It could also be named circle-head.
Enter the radius and angle of the circular segment gap, as well as the height of the rectangle. Choose the number of decimal places and click Calculate.

	Circle radius (r):
	Segment gap angle (α):
	Rectangle height (b):
	Rectangle width (a):
	Arc length (l):
	Total height (h):
	Perimeter (p):
	Area (A):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Radius, width, heights and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

Such a head is not a basic geometric shape but is composed of two such forms, a circular segment, respectively circle and a rectangle. Although this shape lacks a standard name, it, and its three-dimensional equivalent, the ball head, are frequently encountered. In this context, this shape represents the cross-section of the ball head. Such forms are used, for instance, in connectors that stay joined thanks to the enlarged section, preventing them from coming apart on their own. They mate with a corresponding counterpart that shares the same shape as a negative, but is slightly larger. Puzzle pieces are designed similarly in two dimensions, as are ball joints in three dimensions.

This head shape is axially symmetric about the perpendicular bisector of the rectangle's side a. Otherwise it possesses no symmetries. Note that the calculation utilizes the angle α, which is the complement of Θ regarding the circular segment, meaning the two angles together form a full circle of 360 degrees. Calculations within the formulas are performed using radians rather than degrees. In these formulas, h_s represents the height of the circular segment.

Calculation of area and other dimensions: Two-Dimensional Head Shape

Last updated on 08/15/2026. Author: Jürgen Kummer

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