Circles in Geometry
Understanding Circles
A circle is a set of all points in a plane that are equidistant from a fixed point called the center. Circles are fundamental shapes in geometry with unique properties and countless applications.
What You'll Learn:
- Circle definitions and terminology
- Circumference and area formulas
- Arc length and sector area
- Tangent and secant properties
- Circle theorems
- Applications in real life
Circle Terminology
Center
The fixed point from which all points on the circle are equidistant
Radius (r)
The distance from the center to any point on the circle
Diameter (d)
A line segment passing through the center with endpoints on the circle (d = 2r)
Chord
A line segment with both endpoints on the circle
Arc
A portion of the circle's circumference
Sector
Region bounded by two radii and an arc
Essential Formulas
Circumference
The distance around the circle
Area
The region enclosed by the circle
Arc Length
Where θ is the central angle in degrees
Sector Area
Area of a pie-shaped region
Circle Theorems
Tangent Theorem
A tangent to a circle is perpendicular to the radius at the point of tangency
Inscribed Angle Theorem
An inscribed angle is half the central angle that subtends the same arc
Two Tangent Theorem
Two tangent segments from an external point to a circle are equal in length
Practice Problems
Problem 1: Circle Area
Find the area of a circle with radius 7 cm.
Problem 2: Arc Length
Find the arc length of a sector with radius 10 cm and central angle 60°.