Semicircle

In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle.

In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half-disk, which is a two-dimensional geometric region that further includes all the interior points.

A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass.

This can be proven by applying the Pythagorean theorem to three similar right triangles, each having as vertices the point where the perpendicular touches the semicircle and two of the three endpoints of the segments of lengths a and b.

on the diameter between its endpoints and which is entirely concave from below is If it is entirely concave from above, the equation is An arbelos is a region in the plane bounded by three semicircles connected at their endpoints, all on the same side of a straight line (the baseline) that contains their diameters.

Construction of a square with the same area as a given oblong
Proof without words of the AM–GM inequality :
PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b . Using the geometric mean theorem , triangle PGR's altitude GQ is the geometric mean . For any ratio a : b , AO ≥ GQ.
Comparison of Ford circles and a Farey diagram with semicircles for n from 1 to 9. Each semicircle intersects its corresponding circles at right angles. In the SVG image, hover over a circle or curve to highlight it and its terms.
An arbelos (grey region)