Mathematics, Statistics & Geometry

Area of Circular Segment Calculator

Calculate the area of a circular segment (the region between a chord and an arc). Find the chord length, arc length, and segment height instantly.

Segment Area
28.54
Chord Length14.142
Arc Length15.708
Segment Height (h)2.929
Calculation StepsRadius (r) = 10, Angle (θ) = 1.570796 rad Area = 0.5 * r² * (θ - sin(θ)) Area = 0.5 * 10² * (1.570796 - 1.000000) Area = 0.5 * 100 * 0.570796 Area = 28.539816 Chord = 2r * sin(θ/2) = 14.142136 Height = r(1 - cos(θ/2)) = 2.928932

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Advanced Circle Geometry

The Area of Circular Segment calculator is a specialized tool for calculating the area of the "cap" of a circle. This is particularly useful in fluid dynamics (calculating the volume of a partially filled horizontal tank) and structural engineering.

A=12r2(θ−sin⁡θ)\begin{aligned} A = \frac{1}{2} r^2 (\theta - \sin \theta) \end{aligned}

Where:
A=
The area between the chord and the arc
r=
The radius of the circle
θ\theta=
The central angle in radians

Sector vs. Segment: What's the Difference?

  • Sector: Includes the center point of the circle (looks like a slice of pie).
  • Segment: Does not include the center point; it is bounded only by the straight chord and the curved arc.

Key Dimensions Explained

  1. Radius (r): The distance from the circle's center.
  2. Chord (c): The straight line across the circle.
  3. Arc (s): The curved path between the chord's ends.
  4. Height (h): The depth of the segment at its thickest point.

Real-World Use Cases

  • Industrial Engineering: Calculating the volume of liquid in a horizontal cylindrical storage tank.
  • Architecture: Designing arched windows or entryways where the arch is a circular segment.
  • Woodworking: Creating curved table edges or decorative moldings from circular stock.

Frequently Asked Questions

A circular segment is the region of a circle bounded by a chord and an arc. It is essentially a sector with the triangle formed by the radii removed.

The area is calculated by taking the area of the circular sector (0.5 × r² × θ) and subtracting the area of the triangle formed by the two radii and the chord (0.5 × r² × sin θ). This simplifies to 0.5 × r² × (θ - sin θ).

The chord is the straight line connecting the two endpoints of the arc. Its length is calculated as 2r × sin(θ/2).

The height (h) is the perpendicular distance from the midpoint of the chord to the midpoint of the arc. It is calculated as r × (1 - cos(θ/2)).

Yes, the formula is mathematically robust for any angle, including major segments (where θ > 180°).

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