Semicircle

Parts and characteristics

The two primary elements of a semicircle are the arc and the diameter. The arc spans 180 degrees, one half of the full 360° turn of a circle, and every point on the arc is at a constant distance from the circle's center. The diameter is the straight segment connecting the arc's endpoints and passes through the center; it serves as the semicircle's maximal chord. More specific terms include:

• geometric shape boundary: the curved arc plus the diameter segment
• circle relation: a semicircle is exactly one half of a circle
• center and radius: properties inherited from the parent circle

Mathematical properties

Key formulas and facts for a semicircle of radius r include area = 1/2 πr² and boundary length (perimeter) equal to the arc length πr plus the diameter 2r. The arc of a semicircle measures 180° (half of 360 degrees) and is often referred to simply as the arc. A classical result, often taught as Thales' theorem, states that any triangle inscribed in a semicircle with its third vertex on the arc has a right angle at that vertex, so the triangle is a right triangle.

History and context

Semicircles appear early in the development of Greek geometry where properties of circles and inscribed angles were examined. The connection between semicircles and right triangles has been known since antiquity and became a standard tool in classic geometric proofs and constructions.

Uses and examples

Semicircles are common in architecture and engineering (for example, semicircular arches and window designs), in drafting and CAD as primitive shapes, and in mathematical problems ranging from area and arc-length calculations to probability and signal design. They also serve as simple examples when teaching integration, trigonometry, and Euclidean constructions.

Distinctions and notable facts

It helps to distinguish the semicircle (the half-disk region) from a semicircular arc (just the curved boundary) and from an open semicircle where the diameter is not included. The diameter of the semicircle is the longest straight segment inside it and serves as the hypotenuse for any right triangle inscribed with the third vertex on the arc. These elementary properties make the semicircle useful as a teaching and modelling shape in many fields.

Further reading and visual resources can be found via introductory geometry materials and practical design references available online and in textbooks. For related topics see resources on the plane shapes and circle theorems, arc measurement, and constructions with a compass and straightedge.