Circles & Ellipses

Comprehensive collection of geometric calculations for circles, ellipses and related round shapes

Basic Shapes

Circle (A = πr²)
Perfect round shape with constant radius from center point
Ellipse (A = πab)
Oval shape with two foci - basis of conic sections

Circle Arcs & Sectors

Circular Arc (s = rα)
Curved line - part of circle's circumference between two points
Circular Sector (A = ½r²α)
Pie-slice bounded by two radii and an arc
Circular Segment
Cap of a circle - area between chord and circular arc
Circular Angles
Angle relationships in circles - central and inscribed angles

Annular & Hollow Shapes

Annulus (A = π(R² - r²))
Ring between two concentric circles
Annular Sector
Sector of an annulus - like a donut slice

Special Curves & Constructions

Parabolic Arch (y = ax²)
Efficient curve for bridges and arches in structural engineering
Squaring the Circle
Classical problem - square with same area as a circle

About Circle Geometry

Circle geometry forms a fundamental foundation of mathematics and finds practical applications in:

  • Engineering - Gears, bearings
  • Architecture - Arches, domes
  • Astronomy - Planetary orbits
  • Physics - Rotational motion
  • Electronics - Antenna technology
  • Design - Aesthetics and form
Fundamental Circle Formulas
Basic Formulas
Circumference: C = 2πr
Area: A = πr²
Circular Sector
Arc length: s = rα
Area: A = ½r²α
Ellipse
Area: A = πab
Circumference: C ≈ π(a+b)
Annulus
Area: A = π(R² - r²)
Circumference: C = 2π(R + r)
Tip: The mathematical constant π ≈ 3.14159 is the ratio of circumference to diameter and one of the most important constants in mathematics.

Practical Application Examples

Engineering
  • Pipelines: Flow calculations
  • Gears: Transmission ratios
  • Bearings: Load analysis
Architecture
  • Domes: Structural calculations
  • Arches: Load distribution
  • Round buildings: Area planning
Sciences
  • Astronomy: Planetary orbits
  • Physics: Circular motion
  • Optics: Lens calculations
Daily Life & Design
  • Vehicle wheels: Rolling circumference
  • Sports fields: Track calculations
  • Landscaping: Circular garden beds
Quick Reference
πr²
Circle Area
2πr
Circumference
πab
Ellipse
½r²α
Sector
π ≈ 3.14159
Pi Constant
Historical

Archimedes (287-212 BC) calculated π using 96-sided polygons to 3.141...

Squaring the Circle: Attempted since antiquity, proven impossible in 1882.

Modern Computers: π is now known to over 100 trillion digits.