How to Inscribe a Square in any Circle

In the world of geometric construction, inscribing shapes within one another is a classic problem. One such example involves the construction of a square inscribed within a given circle. In this video, Chris Tisdell uses a circle arc template to do so, highlighting the synergy between mathematics and visual artistry.

The objective is to create a square—an equilateral polygon with four equal sides and four right angles—such that it fits perfectly within a predetermined circle. In addition to being a geometrical exercise, it also shows the profound relationship that exist between these fundamental shapes and the mathematical principles that govern their interactions.

Chris Tisdell's method commences with the circle itself, specifically, the given circle within which the square is to be inscribed. The centre of this circle is given, as it serves as a pivotal reference point for the entire construction process. This circle represents what mathematicians call the circumcircle, a circle that encompasses the square and passes through all four of its vertices.

The next step introduces the circle arc template, which plays a central role in guiding the construction.  To begin the construction, Chris Tisdell initiates by drawing a diameter across the circle. This diameter is then perpendicularly bisected to form another diameter that is perpendicular to the first.  The four points of intersection between the diameters and the circle then form the vertices of the square.

At this juncture, the square begins to take shape. By connecting the endpoints of the four line segments, the distinct structure of a square is formed with the straightedge. The beauty of this construction lies in its precision, as the square fits seamlessly within the circle, with its four corners touching the circle's circumference.

A square is characterised by four right angles, and all four sides must be of equal length. Using a protractor, one canconfirm the right angles at each vertex, and a simple measurement ensures thatall four sides are congruent. 

Chris's video showcases the construction of a square inscribed within a given circle, demonstrating an amazing geometrical relationship, which is achieved through the utilisation of a circle arc template.