What are the important properties of circles in geometry?

Key Properties of Circles

• Definition:

A circle is the set of all points in a plane that are a fixed distance (radius) from a central point (center).

• Parts of a Circle:
• radius: distance from center to any point on the circle
• diameter: twice the radius (d = 2r)
• chord: a line connecting two points on the circle
• arc: part of the circumference between two points
• circumference: the total distance around the circle (C = 2πr)
• Important Properties:
• All radii in a circle are equal.
• The diameter is the longest chord.
• Angles subtended by the same arc at the circumference are equal.
• The tangent at any point is perpendicular to the radius at that point.
• The area of a circle is A = πr².

Example:
If a circle has a radius of 4 cm:

• Diameter: d = 2 × 4 = 8 cm
• Circumference: C = 2π × 4 ≈ 25.13 cm
• Area: A = π × 4² ≈ 50.27 cm²

Takeaways:

• Circles are defined by all points at a fixed radius from a center.
• Properties like constant radii, circumferences, and tangents help solve geometric problems.