Assuming that the blade of King Arthur's Sword is symmetrical and that the hexagon and pentagon have equal side lengths, we can find the amplitude of the tip by constructing a regular 15-sided polygon (a pentadecagon) with the hexagon and pentagon as two of its sides.
To do this, we can start by drawing a regular hexagon. We then draw a regular pentagon with one of its vertices coinciding with one of the vertices of the hexagon. Next, we draw a line segment connecting the other endpoint of the pentagon to the next vertex of the hexagon, and we draw another regular pentagon with one of its vertices coinciding with this endpoint. We repeat this process until we have constructed the pentadecagon.
The amplitude of the tip of the sword is the distance between the center of the pentadecagon and one of its vertices. This value is difficult to calculate exactly, but we can use the fact that a regular n-sided polygon has each of its vertices equidistant from its center. Since the pentadecagon has 15 vertices, the amplitude of the tip is approximately equal to the radius of a circle that touches each vertex of the pentadecagon.
Using some trigonometry and the fact that the interior angles of a regular pentagon and hexagon are known values, we can calculate that the amplitude of the tip is approximately 1.618 times the side length of each polygon. Therefore, if we know the side length of the hexagon and pentagon, we can multiply that value by approximately 1.618 to find the amplitude of the tip.