Final answer:
To determine if a quadrilateral is cyclic or circumscribable, it must satisfy certain geometric properties. Arcs are approximated as straight lines when they represent a minor segment of the circle for simplicity in calculations. Comparing geometric shapes helps solve problems related to areas, perimeters, and volumes.
Step-by-step explanation:
To ascertain whether a quadrilateral is cyclic (meaning all its corners lie on a single circle) or circumscribable (meaning a circle can be drawn touching all its sides), there must be specific properties that the quadrilateral satisfies. A quadrilateral is cyclic if the sum of the measures of each pair of opposite angles is 180 degrees, due to the inscribed angle theorem. Additionally, for a quadrilateral to be circumscribable, each pair of consecutive sides must be tangential to an inscribed circle, which means that the sum of one pair of opposite sides must be equal to the sum of the other pair.
Regarding the approximation of arcs, this is done in situations where computing actual arc lengths is complicated and the arc is a small part of the circle. If the side length of a polygon is nearly equal to the arc length for a sufficiently small segment, an approximation is used for simplicity in calculation.
In geometry, setting up problems sometimes necessitates comparing or relating different geometrical shapes and measurements, such as comparing a circle's perimeter to the perimeter of a square that circumscribes it. Understanding these properties allows us to solve problems related to both two-dimensional shapes and three-dimensional objects, including the calculations of areas, volumes, and lengths.