Perimeter of polygons with an inscribed circle

Learning Objectives Define a tangential polygon, an inscribed circle (incircle), and a tangent segment. State and apply the Two Tangent Theorem to find unknown segment lengths. Calculate the length of each side of a tangential polygon by summing its constituent tangent segments. Calculate the total perimeter of a tangential polygon using the lengths of the tangent segments from its vertices. Apply Pitot's Theorem to solve for unknown side lengths in tangential quadrilaterals. Solve multi-step problems involving algebraic expressions for segment lengths in tangential polygons. Ever wondered how a circular gear fits perfectly inside a square casing or why a hexagonal nut fits a wrench so well? ⚙️ This perfect fit is all about the geometry of polygons with inscribed circle...

TermDefinitionExample Inscribed Circle (Incircle)A circle is inscribed in a polygon if every side of the polygon is tangent to the circle. The circle is inside the polygon and touches each side at exactly one point.A circle drawn inside a triangle so that it just touches all three sides. Tangential PolygonA convex polygon that has an inscribed circle. It is also called a circumscribed polygon because it is 'circumscribed about' a circle.A square with a circle inside it that touches the midpoint of all four sides. Point of TangencyThe single point where a side of the polygon touches the inscribed circle.If side AB of a square touches its incircle at point P, then P is the point of tangency. Tangent SegmentThe line segment connecting a vertex of a polygon to one of the two points...

The Two Tangent Theorem If two tangent segments are drawn to a circle from the same external point (a vertex), then they are congruent (equal in length). This is the most important rule for this topic. For any vertex of a tangential polygon, the distances from that vertex to the points of tangency on the two adjacent sides are equal. This property is the key to finding side lengths and perimeters. Perimeter of a Tangential Polygon P = 2(s_1 + s_2 + s_3 + ... + s_n) Where s_1, s_2, ..., s_n are the lengths of the unique tangent segments from each of the n vertices. Since each tangent segment length appears twice around the polygon (once on each adjacent side from a vertex), we can sum the unique segment lengths and multiply by two to find the total perimeter. Pitot's...