A circular segment is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space which is bounded by a circular arc (of less than π radians by convention) and by the circular chord connecting the endpoints of the arc.
The key components are the chord (the straight line forming the base of the segment), the arc (the curved boundary), the central angle (θ) which subtends the arc at the center of the circle, and the height (h) or sagitta, which is the maximum distance from the chord to the arc.
A circle with center O and radius r. A chord, with length c, connects two points on the circumference. This chord and the corresponding arc form the segment. The central angle subtended by the chord at the center is θ. The height of the segment, measured from the midpoint of the chord to the arc, is denoted by h (also called the sagitta). The perpendicular distance from the center O to the chord is the apothem, d.
Symmetry: A circular segment is symmetric about the line that passes through the center of the circle and the midpoint of its chord. This line is perpendicular to the chord.
Relationship to Sector: A segment is intrinsically linked to a circular sector. Its area is derived by subtracting the area of an isosceles triangle (formed by the two radii and the chord) from the area of the sector that shares the same arc and central angle.
Centroid: The center of mass (centroid) of the segment lies on its axis of symmetry. Its distance from the center of the circle depends on the radius and the central angle.
The area of a circular segment can be derived by subtracting the area of the isosceles triangle formed by the radii and the chord from the area of the circular sector defined by the same radii.
The area of a circular sector with radius r and central angle θ (in radians) is given by:
The area of the isosceles triangle with two sides equal to the radius r and the angle between them being θ is given by:
Substituting the formulas for the sector and triangle back into the original relationship gives the final formula for the area of the segment:
Engineers use segment calculations for designing arched supports in bridges and buildings, determining the cross-sectional area of tunnels, and calculating material volumes for curved structures like domes and vaults.
The cross-sectional area of the fluid in a partially filled horizontal cylindrical pipe is a circular segment. This calculation is crucial for determining flow rates, fluid volume, and pressure in pipelines and storage tanks.
The shape of a simple plano-convex or plano-concave lens is based on a circular segment. The geometry of the segment is fundamental to calculating focal length and how the lens refracts light in optical instruments.
In industrial design and manufacturing, segment calculations are used to optimize the cutting of materials from circular stock, minimizing waste. They are also used in designing curved components for machinery and consumer products.
Architectural Arches
Many historical and modern buildings feature arched windows, doorways, and structural supports. These arches are often in the shape of a circular segment, providing both aesthetic appeal and structural strength, as seen in bridges and cathedrals.
Cross-Section of a Pipe
In civil engineering and plumbing, the cross-section of water or other fluids flowing through a partially filled cylindrical pipe forms a circular segment. Calculating its area is crucial for determining flow rates and managing water systems.
Landscape Design
A curved garden bed or patio edge that is bounded by a straight path or wall often forms a circular segment. Landscape architects use this shape to create visually appealing and organic designs in outdoor spaces.
| Type | Central Angle (θ) | Description |
|---|---|---|
| Minor Segment | θ < 180° (π rad) | The smaller of the two segments created by a chord. |
| Major Segment | θ > 180° (π rad) | The larger of the two segments created by a chord. |
| Semicircle | θ = 180° (π rad) | A special case where the chord is the diameter of the circle. |
| Quarter-Circle Segment | θ = 90° (π/2 rad) | The segment formed by a chord connecting the endpoints of two perpendicular radii. |
Using degrees in the radian formula. The formula A = (1/2)r²(θ - sinθ) is only valid when θ is in radians. Always convert angles to radians first.
Confusing segment with sector. A sector is a pie-slice shape bounded by two radii and an arc. A segment is bounded by a chord and an arc. Remember to subtract the triangle from the sector.
Incorrectly calculating the sine of the angle. Ensure your calculator is in the correct mode (radians or degrees) to match the angle unit you are using for the sine function.