Circles are fascinating geometric shapes with unique properties. They're defined by a center point and a constant radius, which creates a perfectly round shape. Understanding circles is crucial for grasping many geometric concepts and real-world applications.

Circle properties include radii, diameters, chords, and tangent lines. These elements help us calculate important measurements like circumference and area. We also explore relationships between inscribed and circumscribed figures, which connect circles to other polygons in interesting ways.

Circle Properties and Relationships

Parts of a circle

• Center: point equidistant from all points on the circle
• Radius: line segment from the center to any point on the circle
  • All radii of a circle are congruent
• Diameter: line segment that passes through the center and has its endpoints on the circle
  • Diameters are the longest chords in a circle (chord: line segment with both endpoints on the circle)
• Secant: line that intersects a circle at two points
• Tangent: line that intersects a circle at exactly one point, called the point of tangency
  • Tangent lines are perpendicular to the radius drawn to the point of tangency

Radius vs diameter relationship

• Diameter of a circle is twice the length of its radius
  • $diameter = 2 \times radius$ or $d = 2r$
• Radius of a circle is half the length of its diameter
  • $radius = \frac{diameter}{2}$ or $r = \frac{d}{2}$

Circumference and area calculations

• Circumference: distance around a circle
  • Formula: $C = 2\pi r$ or $C = \pi d$, where $\pi \approx 3.14159$
• Area: space enclosed by a circle
  • Formula: $A = \pi r^2$
• Arc length: length of an arc, a portion of the circumference
  • Formula: $arc\ length = \frac{central\ angle\ in\ degrees}{360^\circ} \times 2\pi r$
• Sector area: area of a sector, a portion of the circle's area
  • Formula: $sector\ area = \frac{central\ angle\ in\ degrees}{360^\circ} \times \pi r^2$

Inscribed and circumscribed figures

• Inscribed figures: polygon with all vertices on the circle
  • Center of an inscribed circle is the intersection of the polygon's angle bisectors
  • Radius of an inscribed circle is perpendicular to each side of the polygon at the point of tangency
• Circumscribed figures: polygon with each side tangent to the circle
  • Center of a circumscribed circle is the intersection of the perpendicular bisectors of the polygon's sides
  • Radius of a circumscribed circle is perpendicular to each side of the polygon at the point of tangency
• Inscribed angles: angle formed by two chords with a common endpoint on the circle
  • Measure of an inscribed angle is half the measure of its central angle
  • Inscribed angles subtended by the same arc are congruent (arc: portion of the circle between two points)