Overview
This lecture introduces the basic properties of circles, important formulas, and applications such as finding area, circumference, sector area, arc length, and relationships involving central and inscribed angles.
Circle Basics
- The radius is a segment from the center of the circle to any point on the circle.
- The diameter passes through the center and is twice the radius.
- The area of a circle is ( \pi r^2 ).
- The circumference (perimeter) of a circle is ( 2\pi r ) or ( \pi d ), where ( d ) is the diameter.
Sectors and Arcs
- A sector is a portion of a circle defined by a central angle.
- The area of a sector is ( \frac{\theta}{360} \times \pi r^2 ), where ( \theta ) is the angle in degrees.
- The arc length is ( \frac{\theta}{360} \times 2\pi r ).
Chords, Central, and Inscribed Angles
- A chord is a line segment connecting any two points on a circle.
- The diameter is a special chord passing through the center.
- A central angle is formed at the center and its intercepted arc equals the angle.
- An inscribed angle is formed on the circle; its intercepted arc is twice the angle.
- If an inscribed angle is ( x ) degrees, its arc is ( 2x ) degrees; if a central angle is ( x ) degrees, its arc is also ( x ) degrees.
Example Problems
- Given diameter ( d = 8 ) cm: circumference ( = 8\pi ) cm (( \approx 25.1 )), area ( = 16\pi ) cm² (( \approx 50.27 )).
- Given area ( = 81\pi ), radius ( = 9 ), circumference ( = 18\pi ) (( \approx 56.55 )), diameter ( = 18 ).
- For circle with radius 7 in., central angle 150°: arc length ( = \frac{35\pi}{6} ) (( \approx 18.3 )), sector area ( = \frac{245\pi}{12} ) (( \approx 64.1 )).
- For values of ( x ) and ( y ) in figures, use central and inscribed angle relationships.
- For shaded region in circle with ( AB = 48 ), ( BC = 14 ): radius ( = 25 ), area shaded ( = 625\pi - 336 ) (( \approx 1278.5 )).
Key Terms & Definitions
- Radius — line from center to circle edge.
- Diameter — line through center; twice the radius.
- Circumference — perimeter of the circle.
- Sector — region bounded by two radii and their arc.
- Arc — a part of the circle's edge.
- Chord — segment joining two circle points.
- Central Angle — angle with vertex at circle's center.
- Inscribed Angle — angle with vertex on the circle.
Action Items / Next Steps
- Practice calculating area, circumference, arc length, and sector area for different circles.
- Review relationships between angles and arcs in circle diagrams.
- Solve additional problems involving chords, radii, and diameters.