Calculate radius, diameter, circumference, and area of a circle. Enter any one value to automatically calculate all other circle properties with step-by-step solutions and visual representation.
A circle is one of the most fundamental geometric shapes, defined as the set of all points in a plane that are equidistant from a given center point. Understanding circle properties and relationships is essential for geometry, trigonometry, engineering, and many real-world applications.
Circle calculations are used in architecture, engineering design, manufacturing, astronomy, and countless other fields where circular shapes and rotational motion are important. Mastering circle formulas helps solve problems involving wheels, gears, orbits, and circular structures.
Diameter is twice the radius
The distance around the circle
The space enclosed by the circle
| Application | Circle Property | Usage | Examples |
|---|---|---|---|
| Wheel Design | Circumference | Distance calculations | Vehicle wheels, bicycle tires |
| Construction | Area | Material planning | Circular foundations, domes |
| Manufacturing | Diameter | Tolerance specifications | Pipes, rods, circular parts |
| Astronomy | Circumference | Orbital calculations | Planetary orbits, satellite paths |
| Sports | Radius | Field design | Athletic tracks, center circles |
| Architecture | Area | Space planning | Circular rooms, rotundas |
| Engineering | All properties | Design calculations | Gears, pulleys, turbines |
Definition: Pi (Ï€) is the ratio of a circle's circumference to its diameter. This ratio is constant for all circles, regardless of size.
Value: π ≈ 3.14159265... Pi is an irrational number, meaning its decimal representation never ends or repeats in a pattern.
History: Pi has been studied for over 4,000 years, with ancient civilizations approximating its value. Modern computers have calculated π to trillions of decimal places.
Applications: Pi appears in many mathematical formulas beyond circle geometry, including statistics, physics, and engineering equations.
Chord Properties: A chord is a line segment connecting two points on a circle. The perpendicular from the center to a chord bisects the chord.
Tangent Properties: A tangent line touches the circle at exactly one point and is perpendicular to the radius at that point.
Sector and Segment: A sector is a "pie slice" of a circle, while a segment is the region between a chord and the arc it cuts off.
Inscribed Angles: An angle inscribed in a circle is half the central angle that subtends the same arc.
| Measurement | Linear Units | Area Units | Common Usage |
|---|---|---|---|
| Radius/Diameter | mm, cm, m, km, in, ft | - | Length measurements |
| Circumference | mm, cm, m, km, in, ft | - | Perimeter measurements |
| Area | - | mm², cm², m², km², in², ft² | Surface area calculations |
| Arc Length | mm, cm, m, km, in, ft | - | Partial circumference |
| Angular Measure | degrees (°), radians | - | Angle measurements |
Radian Measure: An alternative to degrees for measuring angles. One radian is the angle subtended by an arc equal in length to the radius.
Arc Length: The distance along the circumference between two points. Calculated as s = rθ, where θ is in radians.
Sector Area: The area of a "pie slice" section. Calculated as A = ½r²θ for angle θ in radians.
Circle Equations: In coordinate geometry, circles can be described by equations like (x-h)² + (y-k)² = r², where (h,k) is the center.
Engineering Applications: Calculate material needed for circular components, determine gear ratios, design circular foundations and structures.
Navigation and Mapping: Use circle properties to calculate distances, plot circular routes, and determine coverage areas for signals or services.
Manufacturing: Ensure proper dimensions for circular parts, calculate tolerances, and optimize material usage in circular designs.
Science and Research: Model planetary motions, design experiments involving rotation, and analyze data with circular patterns.