Inscribed and circumscribed figures

Inscribed and circumscribed figures describe complementary ways in which rectilinear figures and circles are fitted together. An inscribed figure is governed by the positions of its vertices on a containing boundary, whereas a circumscribed figure is governed by sides or lines that pass through, or touch, the boundary of another figure. In the geometry of Euclid, these are not merely visual arrangements: they are defined conditions that make later constructions and demonstrations precise.

Definitions and Geometric Boundaries

A rectilinear figure is said to be inscribed in another when all the angular points of the inscribed figure lie on the containing figure. Applied specifically to a circle, the definition becomes sharper: a rectilinear figure is inscribed in a circle when the vertex of each of its angles is on the circumference. The polygon is therefore contained by the circle while its corners are fixed upon the circular boundary.

The reverse arrangement is described through the sides of the outer figure. A figure is described about another when all the sides of the circumscribed figure pass through the angular points of the inner figure. For a circle, however, the decisive condition is tangency: a rectilinear figure is circumscribed about a circle when each of its sides is a tangent to the circle. The circle is then enclosed by straight sides, each touching it without cutting through it.

The definitions also reverse the grammatical direction of the relation. A circle is inscribed in a rectilinear figure when every side of the figure is a tangent to the circle. A circle is circumscribed about a rectilinear figure when its circumference passes through the vertex of each angle of that figure. Thus the same configuration can be described from either figure's point of view: a polygon may be inscribed in a circle, and the circle may be circumscribed about that polygon; a circle may be inscribed in a polygon, and the polygon may be circumscribed about the circle.

Construction and Proof

The concept belongs to a geometry whose permitted instruments are deliberately restricted. The straight-edge and compasses are identified as the only instruments allowed in Euclid's plain geometry, and the postulates exist in part to declare this limitation. A construction of an inscribed or circumscribed figure must therefore be expressed through operations available to those instruments rather than through an unexamined appeal to measurement or mechanical drawing.

That restriction separates the visible act from the mathematical claim. A problem proposes that something be done, such as drawing a line, describing a circle, or constructing a figure; its solution shows how the required object can be made with the straight-edge and compasses. The demonstration then proves that the indicated process actually reaches the required end. In this framework, placing vertices on a circumference or making sides tangent to a circle is not established solely because a diagram appears to show it. The construction must be connected to definitions, postulates, axioms, and previously proved propositions.

The general Euclidean system begins with definitions and grants that a straight line may be drawn from any one point to any other point, while a circle may be described with any centre at any distance from that centre. These permissions make the two boundaries central to the subject possible: the straight lines that form polygonal sides and the circles that provide circumferences, centres, and radii. Superposition, defined as conceiving one magnitude placed upon another so that every part exactly coincides, supplies another mode of exact comparison within the same system.

Applications in Book IV

The definitions become practical in Book IV, where the text treats the relations between polygons and circles as construction problems rather than as isolated vocabulary. One such problem is to inscribe a circle in a given square. The title itself establishes the required direction of the operation: the square is the enclosing rectilinear figure, and the circle must be placed within it so that each side of the square is tangent to the circle. The problem therefore turns the definition of an inscribed circle into a concrete demand for a construction and a proof of contact on all four sides.

A later problem asks for a circle to be described about a given equilateral and equiangular pentagon. Here the pentagon is the inner figure and the circle must pass through the vertex of each angle. The construction consequently seeks the circular boundary that recognizes all five vertices at once. The pentagon's equilateral and equiangular character matters because the figure supplies a highly ordered set of vertices, while the required circle converts that order into a common circumference.

These two problems stage the concept in opposite directions. Inscribing a circle in a square works inward from straight sides toward a tangent circular boundary; describing a circle about a regular pentagon works outward from polygonal vertices toward a circumference passing through them. Together they show why the definitions distinguish sides, vertices, tangency, and circumference so carefully: each condition controls a different kind of fit.

Relation to Euclidean Method

Inscribed and circumscribed figures reveal how Euclidean geometry coordinates visual intuition with formal reasoning. The work acknowledges that instruction can become dry and rigid when demonstrations are buried beneath needless repetition, but it presents geometry as a discipline capable of awakening curiosity and strengthening the reasoning faculties. Its colored and visible figures are intended to appeal directly to the eye, while the definitions prevent visual immediacy from becoming vagueness.

The resulting method is both pictorial and axiomatic. A reader can see a polygon inside a circle or a circle touching the sides of a polygon, yet the text specifies exactly what must be seen and what must be proved: every relevant vertex lies on the circumference, or every relevant side is a tangent. The visual configuration supplies access to the idea; the construction and demonstration establish its validity. In this sense, inscribed and circumscribed figures exemplify the larger educational purpose of the book, which seeks to teach people how to think rather than merely what to think.