4.4.6 Circles and Polygons¶

Exploring inscribed and circumscribed polygons, and the relationships between circles and regular polygons.

定义¶

An inscribed polygon is a polygon whose vertices all lie on a circle (called the circumscribed circle or circumcircle). A circumscribed polygon is a polygon whose sides are all tangent to a circle (called the inscribed circle or incircle). For a regular polygon with \(n\) sides inscribed in a circle of radius \(R\), the polygon has equal side lengths and equal interior angles. The relationship between a regular polygon and its circumscribed or inscribed circle involves the apothem (the perpendicular distance from the center to a side) and the radius of the circle. Key properties include: (1) All vertices of an inscribed polygon lie on the circumcircle; (2) All sides of a circumscribed polygon are tangent to the incircle; (3) For a regular polygon, the center of the inscribed and circumscribed circles coincide; (4) The angle subtended by each side at the center equals \(\frac{2\pi}{n}\) radians or \(\frac{360°}{n}\) for an \(n\)-sided regular polygon.

核心公式¶

\(a = R\cos\left(\frac{\pi}{n}\right)\)
\(s = 2R\sin\left(\frac{\pi}{n}\right)\)
\(A_{polygon} = \frac{1}{2}nr^2\tan\left(\frac{\pi}{n}\right) = \frac{1}{2}nRs\)
\(r = R\cos\left(\frac{\pi}{n}\right)\)
\(P = 2nR\sin\left(\frac{\pi}{n}\right)\)

易错点¶

⚠️ Confusing the apothem of an inscribed polygon with the radius of the circumscribed circle—the apothem is the distance from the center to the midpoint of a side, not to a vertex
⚠️ Incorrectly applying the relationship between inscribed and circumscribed circles; for an inscribed polygon the radius \(R\) refers to the circumcircle, while for a circumscribed polygon the radius \(r\) refers to the incircle
⚠️ Forgetting to use the correct angle \(\frac{\pi}{n}\) or \(\frac{180°}{n}\) in formulas; students often mistakenly use \(\frac{2\pi}{n}\) (the full central angle) instead of half that angle in apothem calculations
⚠️ Mixing up the formulas for area and perimeter, or applying formulas for regular polygons to irregular inscribed/circumscribed polygons without recognizing that the polygon must be regular for these standard formulas to apply