Circle Segment Calculator

What is a Circle Segment?

A circle segment is the region of a circle bounded by a chord and the arc subtended by that chord. Unlike a sector (which is bounded by two radii and an arc), a segment is defined by a straight line cutting across the circle. Circle segments appear frequently in engineering, architecture, and physics when calculating cross-sections of pipes, designing arches, or analyzing partially filled containers.

Key Measurements of a Circle Segment

A circular segment is described by these fundamental measurements:

Radius (r) - The distance from the center of the circle to its circumference.
Central Angle (θ) - The angle subtended by the chord at the center of the circle.
Segment Height / Sagitta (h) - The perpendicular distance from the midpoint of the chord to the arc.
Chord Length (c) - The straight line distance between the two endpoints of the arc.
Arc Length (s) - The curved distance along the circumference between the chord endpoints.
Segment Area (A) - The area enclosed by the chord and the arc.

How to Use the Circle Segment Calculator

This calculator supports two modes of calculation based on the inputs you have available:

Radius and Central Angle

When you know the radius of the circle and the central angle (in degrees or radians), the calculator computes the segment area, chord length, arc length, and sagitta. Use this mode when you have angle measurements from a protractor or design specification.

Radius and Segment Height

When you know the radius and the sagitta (segment height), the calculator determines the central angle, segment area, chord length, and arc length. This mode is useful for real-world measurements where you can measure the height of a segment directly but not the angle.

Circle Segment Formulas

The calculations use these standard geometric formulas:

Arc Length: s = r × θ (θ in radians)
Chord Length: c = 2r × sin(θ/2)
Segment Height: h = r × (1 - cos(θ/2))
Segment Area: A = (r²/2) × (θ - sin θ)
Central Angle from Height: θ = 2 × arccos(1 - h/r)

The segment area formula derives from subtracting the area of the isosceles triangle (formed by the two radii and chord) from the sector area. The result is the area of the region between the chord and the arc.

Practical Applications

Circle segment calculations are used in many fields. Engineers calculate the cross-sectional area of partially filled pipes and tanks. Architects design arches and curved structures using segment geometry. Woodworkers and metal fabricators use segment measurements when cutting curved pieces. Understanding segment properties is also essential in optics for lens design and in civil engineering for tunnel cross-sections.

Frequently Asked Questions

What is the difference between a circle sector and a circle segment?

A sector is a region bounded by two radii and an arc (like a pizza slice). A segment is bounded by a chord and the arc above it (like a slice without the tip). The segment area equals the sector area minus the area of the triangular region formed by the two radii and the chord.

What is sagitta in a circle segment?

The sagitta, also called the segment height, is the perpendicular distance from the midpoint of the chord to the arc. It represents the depth or rise of the segment. The term comes from the Latin word for "arrow," describing the arrow-like shape formed by the chord and arc.

Can the central angle be greater than 180 degrees?

Yes. When the central angle exceeds 180 degrees, you get a major segment (larger than a semicircle). The formulas work correctly for all angles up to 360 degrees. For angles over 180 degrees, the segment area will be larger than half the circle area.

How do I find the radius if I know the chord length and sagitta?

You can calculate the radius using the formula: r = (c² / 8h) + (h / 2), where c is the chord length and h is the sagitta. This is useful when you can only measure the chord and the depth of a segment directly.

What are typical real-world examples of circle segments?

Common examples include the cross-section of a partially filled pipe or tank, the shape of an arch above a doorway, the lens of eyeglasses, and the cross-section of a tunnel. Circle segments also appear in gear teeth design, bridge engineering, and decorative architectural elements.

Why is the segment area formula called transcendental?

The segment area formula involves the inverse cosine (arccos) function, which is a transcendental function. This means there is no closed-form algebraic solution to solve for the radius given only the area and height. Numerical methods or iterative calculations are required for such inverse problems.

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