Segment Area Calculator

Calculate circular segment area, chord length, arc length, central angle, and perimeter based on radius and segment height. Segment Area Calculator helps students, engineers, designers, and construction professionals analyze circular sections accurately.

Highlights

  • Calculate circular segment area from radius and segment height.
  • Find chord length, arc length, and central angle instantly.
  • Determine the total perimeter of a circular segment.
  • Supports multiple metric and imperial measurement units.
  • Useful for students, engineers, designers, and construction professionals.

Table of Contents

  1. How Segment Area Calculator Works?
  2. Practical Use Cases
  3. Accuracy and Limitations
  4. Quick Examples
  5. Frequently Asked Questions (FAQs)
  6. Cite This Calculator

How Segment Area Calculator Works?

A circular segment is the region enclosed by a chord and the corresponding part of a circle's arc. This calculator uses the circle radius r and the perpendicular segment height, or sagitta, h, to determine the segment area and related dimensions. For a valid circular segment, the height must satisfy 0 < h ≤ 2r.

First, the central angle θ is calculated in radians from the radius and segment height:

θ=2cos-1(r-hr)

The same angle in degrees is found using:

θdeg=θ×180π

The straight chord connecting the two endpoints of the arc is:

c=22rh-h2

The curved arc length is calculated from the radius and the central angle in radians:

L=rθ

The circular segment area is the area of the sector minus the area of the associated triangle:

A=12r2(θ-sinθ)

Finally, the segment perimeter consists of the curved arc and the straight chord:

P=L+c

Calculated resultMeaning
Segment AreaArea between the circular arc and chord
Chord LengthStraight-line distance between the arc endpoints
Arc LengthCurved distance along the circle
Central AngleAngle at the circle center, reported in degrees or radians
Segment PerimeterArc length plus chord length

The calculator supports radius and segment-height inputs expressed in mm, cm, m, km, in, ft, or yd. Length-based results use the corresponding squared unit for area, while the central angle is available in degrees or radians.

Practical Use Cases

Students can use the calculator to check geometry exercises involving circular segments, including the relationship between radius, sagitta, chord length, sector angle, and area. It provides several calculated values that help verify work from trigonometric and circle-geometry formulas.

Engineers can estimate the area and boundary dimensions of curved sections in mechanical components, pressure vessels, structural profiles, and other designs that include a circular arc joined to a chord.

Designers can determine the dimensions of arched panels, rounded openings, furniture profiles, logos, and other visual elements. The chord and arc outputs help with layout, while the area result supports material and surface planning.

Construction professionals can analyze curved openings, arch sections, segmented ceilings, paving layouts, and formwork. The segment perimeter helps estimate the total boundary length, and the area helps quantify the enclosed section.

Accuracy and Limitations

The calculator uses standard circular-segment equations and provides accurate numerical results for positive radius values and geometrically valid heights from just above zero through 2r. The height must not exceed twice the radius, because that would not define a chord-based circular segment. Results near the limiting values h = 0 or h = 2r can be more sensitive to numerical rounding, especially for very small segments. The calculation assumes a perfect planar circle, an exact perpendicular segment height, and no manufacturing, material-thickness, or measurement tolerances; it also does not account for irregular curves or three-dimensional surfaces.

Quick Examples

Example #1: Small Circular Segment

Calculate the dimensions and area of a shallow circular segment for a small design or geometry exercise.

Example #2: Large Architectural Segment

Determine the area, chord, arc, and perimeter of a circular segment used in an architectural or construction layout.

Example #3: Semicircular Segment

Analyze a segment with a height equal to its radius, representing a semicircular segment configuration.

Frequently Asked Questions

A circular segment is the region enclosed by a chord and the corresponding arc of a circle. It is different from a sector, which is bounded by two radii and an arc.

Segment height is the perpendicular distance from the chord to the circular arc. It is also called the sagitta.

Chord length is the straight-line distance between two points on a circle. Arc length is the distance measured along the curved portion of the circle between those same points.

They are useful for estimating curved areas, material quantities, structural dimensions, and boundary lengths in fields such as construction, engineering, design, and surveying.

Use the same unit for both measurements to maintain consistency. The resulting area will be in squared units, while lengths will remain in the selected linear unit.

Cite This Calculator