Circle Measurements Calculator

Complete circle measurement tool. Enter the radius and an optional angle to compute all circle properties.

Enter Circle Dimensions

Result

Circle Area
314.159265
square units
Radius10
Diameter20
Circumference62.831853
Area314.159265
Sector Area (60°)52.359878
Arc Length (60°)10.471976
Segment Area (60°)9.058040
Chord Length (90°)14.142136

Step-by-Step Solution

A = πr² = π(10)² = 314.159265

Complete Circle Measurements Guide

A circle is fully defined by a single measurement -- its radius. From the radius alone, every other property of the circle can be derived. This calculator computes all major circle measurements including area, circumference, diameter, sector area, segment area, and chord length.

Circle Measurement Formulas

Area

The total space enclosed within the circle boundary.

A = πr²

Circumference

The total distance around the circle perimeter.

C = 2πr

Diameter

The longest distance across the circle through the center.

d = 2r

Sector Area

The area of a pie-shaped slice defined by a central angle.

A = (θ/360)πr²

Segment Area

The area between a chord and the arc it subtends.

A = (r²/2)(θ - sinθ)

Chord Length

The straight-line distance between two points on the circle.

c = 2r sin(θ/2)

Understanding Sectors and Segments

Sector

A sector is a "pie slice" of a circle bounded by two radii and the arc between them. Its area is a fraction of the total circle area, proportional to the central angle. For a sector with central angle θ (in degrees): Sector Area = (θ/360) x πr². The arc length of the sector boundary is Arc = (θ/360) x 2πr.

Segment

A circular segment is the region between a chord and the arc it cuts off. To find its area, subtract the area of the triangle formed by the two radii and the chord from the sector area. Using the angle in radians: Segment Area = (r²/2)(θ - sinθ). This formula works for angles from 0 to 2π radians (0° to 360°).

Chord

A chord is a line segment whose endpoints lie on the circle. The longest possible chord is the diameter. The chord length for a given central angle θ is: Chord = 2r sin(θ/2). As the angle increases from 0 to 180°, the chord length increases from 0 to the diameter (2r). Beyond 180°, the chord length decreases symmetrically.

Practical Applications

  • Architecture: Sector areas are used for designing arched windows, fan-shaped rooms, and circular driveways.
  • Engineering: Segment areas help calculate the cross-sectional area of partially filled pipes and tanks.
  • Navigation: Chord length calculations are essential in radar systems and satellite positioning.
  • Manufacturing: Cutting circular materials into sectors for cones, fans, and packaging requires precise angle-based calculations.
  • Sports: Throwing sectors in discus, shot put, and javelin events are defined by sector geometry.

Key Relationships

  • A semicircle (180°) has a sector area of πr²/2 and chord length equal to the diameter 2r.
  • A quarter circle (90°) has a sector area of πr²/4 and chord length of r√2.
  • A 60° sector forms an equilateral triangle with the two radii and the chord (since chord = r when θ = 60°).
  • The segment area approaches zero for very small angles and approaches the full circle area for angles near 360°.