Calculations for a rectangular star. Such a rectangular star, featuring n rectangular spikes, is formed by taking a regular polygon with 2n vertices and extending two adjacent vertices inward toward the center, perpendicular to the connecting edge and parallel to one another. This process is repeated with the next pair of vertices, and so on, all the way around the figure. The rectangles extend just far enough to touch their adjacent rectangles. The result is a set of identical, circumferentially arranged rectangles surrounding a central regular polygon with n vertices and an edge length of a.
Enter width of a rectangle and the number of spikes n≥3. Choose the number of decimal places. Then click Calculate.
Formulas:
Lengths and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
In two dimensions, spikes typically refer to sharp vertices, specifically, two straight lines intersecting at an acute angle. However, this is not the intended meaning of spikes in this context. Here, the spikes are the segments projecting outward. These segments have a width of a and the vertices are spaced equidistantly at their adjacent vertices, as the vertices of the surrounding polygon are regularly distributed. The gaps between them are isosceles triangles with a base of length a and legs of length b. The angle between these sides is 360°/n. Consequently, this shape can also be conceptualized as a regular polygon with 2n vertices, from which n such isosceles triangles have been removed, one from every second edge. A four-pointed rectangular star constitutes a specific type of cross. However, in the case of the cross versus the rectangular star, the variables a and b are assigned in reverse order.
The circumcircle originates from the center and touches all the outward-pointing vertices of the rectangular star. It is the same circumcircle as that of the surrounding regular polygon with 2n vertices. The inward-pointing vertices have an angle twice that formed by two lines extending from adjacent vertices of the surrounding polygon to the center.