Segment Area Calculator
Solve segment area problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Segment Area Calculator
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Formula: A = (r^2 / 2)(theta - sin(theta))
Worked example — Segment area = 28.5398 sq units | Chord = 14.1421 | Arc = 15.7080
Formula
A = (r^2 / 2)(theta - sin(theta))
Where A is the segment area, r is the radius, and theta is the central angle in radians. This equals the sector area minus the triangle area. The sector area is (1/2)r^2 theta and the triangle area is (1/2)r^2 sin(theta).
Worked Examples
Example 1: Segment Area with 90-Degree Central Angle
Problem:Find the area of a circular segment with radius 10 and central angle 90 degrees.
Solution:theta = 90 degrees = pi/2 = 1.5708 radians Sector area = (1/2)(10^2)(1.5708) = 78.5398 Triangle area = (1/2)(10^2)(sin 90) = 50.0000 Segment area = 78.5398 - 50.0000 = 28.5398 Chord length = 2(10)sin(45) = 14.1421 Arc length = 10(1.5708) = 15.7080
Result:Segment area = 28.5398 sq units | Chord = 14.1421 | Arc = 15.7080
Example 2: Semicircular Segment (180 Degrees)
Problem:Find the segment area for a circle with radius 5 and central angle 180 degrees.
Solution:theta = 180 degrees = pi radians Sector area = (1/2)(25)(pi) = 39.2699 Triangle area = (1/2)(25)(sin 180) = 0 Segment area = 39.2699 - 0 = 39.2699 This equals pi*r^2/2 = 39.2699 (semicircle) Chord = 2(5)sin(90) = 10 (diameter)
Result:Segment area = 39.2699 (semicircle) | Chord = 10.0000 (diameter)
Frequently Asked Questions
What is a circular segment and how is its area calculated?
A circular segment is the region between a chord and the arc it subtends on a circle. It is essentially a slice of a circle with the triangular part removed. The area is calculated using the formula A = (r^2/2)(theta - sin(theta)), where r is the radius and theta is the central angle in radians. This formula works by computing the area of the circular sector (the pie-slice shape from the center) and subtracting the area of the isosceles triangle formed by the two radii and the chord. The sector area is (1/2)r^2 theta and the triangle area is (1/2)r^2 sin(theta). The segment area equals their difference. For small angles, the segment area approaches zero, while for an angle of 180 degrees (pi radians), the segment equals a semicircle.
What is the difference between a circular segment and a circular sector?
A circular sector and a circular segment are distinct geometric regions. A sector is the pie-shaped region bounded by two radii and an arc, like a slice of pizza. Its area is (1/2)r^2 theta. A segment is the region between a chord and the arc it cuts off, like the shape you get when you cut straight across a circle. The segment area equals the sector area minus the triangle area formed by the two radii and the chord. When the central angle is less than 180 degrees, the segment is called a minor segment, and the remaining larger region is the major segment. The sector always includes the center of the circle, while the segment never does (unless the angle is exactly 360 degrees). Understanding this distinction is important for correctly computing areas in engineering and design.
What is the sagitta and how does it relate to the segment?
The sagitta (also called the height of the segment) is the perpendicular distance from the midpoint of the chord to the arc. It is calculated as h = r(1 - cos(theta/2)), where r is the radius and theta is the central angle. The sagitta is the maximum height of the segment and is an important measurement in optics, architecture, and engineering. In lens design, the sagitta determines the curvature of lens surfaces. In architecture, it describes the rise of an arch. The sagitta can also be used to find the radius of a circle when the chord length and sagitta are known: r = (h/2) + (c^2)/(8h), where c is the chord length. This reverse calculation is particularly useful in field measurements where the radius cannot be directly measured.
How do you calculate the arc length and chord length of a segment?
The arc length is the curved distance along the circumference from one end of the chord to the other, calculated as L = r theta, where theta is in radians. For degrees, use L = (theta/360) times 2 pi r. The chord length is the straight-line distance between the two endpoints, calculated as c = 2r sin(theta/2). These two measurements, along with the segment area, completely describe the geometric properties of the segment. The ratio of arc length to chord length is always greater than 1 for non-zero angles and approaches 1 as the angle approaches zero. For a semicircle (theta = 180 degrees), the arc length is pi r and the chord length is 2r (the diameter), giving a ratio of pi/2 which is approximately 1.5708. These calculations are essential in civil engineering for designing curved roads and bridges.
When is a segment a minor segment versus a major segment?
A minor segment is created when the central angle is less than 180 degrees (pi radians). It is the smaller region between the chord and the shorter arc. A major segment is created when the central angle exceeds 180 degrees, and it represents the larger region. Alternatively, the major segment is the complement of the minor segment with respect to the full circle: major segment area = circle area minus minor segment area. When the central angle is exactly 180 degrees, the chord is a diameter, and both segments are equal semicircles. Segment Area Calculator computes the minor segment by default and also shows the major segment area. In applications like water in a cylindrical tank lying horizontally, the cross-section of water forms a segment, and knowing whether it is minor or major determines which formula gives the correct volume.
How is the segment area formula derived step by step?
The derivation begins with the circular sector, which is a fraction of the total circle. The sector area equals (theta / (2 pi)) times pi r^2 = (1/2)r^2 theta. Next, we compute the area of the isosceles triangle formed by the two radii and the chord. Using the formula for triangle area with two sides and the included angle: triangle area = (1/2)r times r times sin(theta) = (1/2)r^2 sin(theta). The segment is what remains after removing the triangle from the sector, so segment area = (1/2)r^2 theta - (1/2)r^2 sin(theta) = (1/2)r^2(theta - sin(theta)). This derivation assumes theta is in radians. The formula can also be derived using integration in polar coordinates, integrating the area between the chord (expressed as a line in polar form) and the arc from one intersection point to the other.
What practical applications use circular segment area calculations?
Circular segment areas appear in numerous engineering and scientific applications. In hydraulics, the cross-sectional area of flow in a partially filled pipe is a circular segment, essential for calculating flow rates using the Manning equation. In architecture, arched windows and doorways create segment-shaped openings whose area must be computed for glazing and structural loads. In manufacturing, when a cylindrical rod is cut by a flat plane, the cross-section is a segment, important for material waste calculations. In agriculture, the coverage area of a center-pivot irrigation system that cannot complete a full rotation involves segment geometry. In astronomy, the illuminated portion of the Moon during phases forms a lens shape composed of two circular segments. These diverse applications demonstrate why segment area calculation is a fundamental skill in applied mathematics.
How does the segment area change as the central angle varies?
The relationship between central angle and segment area is nonlinear. At very small angles, the segment area is approximately (r^2 theta^3) / 12, growing as the cube of the angle, meaning it starts very small. As the angle increases toward 90 degrees, the segment becomes a more significant portion of the circle. At 90 degrees, the segment area is r^2(pi/4 - 1/2), approximately 0.2854 r^2. At 180 degrees (semicircle), the segment area is (pi/2)r^2, exactly half the circle, because sin(pi) = 0 so the triangle area is zero. Between 180 and 360 degrees, the segment grows to encompass most of the circle, reaching the full circle area at 360 degrees. The growth rate is fastest around 180 degrees where d(area)/d(theta) = (r^2/2)(1 - cos(theta)) is maximized at theta = pi.
Can you calculate the radius from a known chord length and sagitta?
Yes, the radius can be determined from the chord length and sagitta using the formula r = (h/2) + (c^2)/(8h), where h is the sagitta (height) and c is the chord length. This formula is derived from the relationship between the radius, half-chord, and the apothem: r^2 = (c/2)^2 + (r - h)^2. Expanding and solving for r gives the formula above. This reverse calculation is extremely useful in field surveying when you can measure the chord and sagitta of an existing curved structure but cannot determine the center or radius directly. Once the radius is known, the central angle can be found using theta = 2 arcsin(c/(2r)), and then all other segment properties can be computed. This technique has been used by architects and engineers since antiquity to reconstruct curved designs from partial measurements.
How do you calculate the volume of a partially filled horizontal cylinder?
A horizontal cylindrical tank partially filled with liquid has a cross-section that is a circular segment. The volume of liquid equals the segment area multiplied by the length of the cylinder: V = L times (r^2/2)(theta - sin(theta)), where L is the cylinder length. To use this formula, you first need theta, which can be found from the liquid height h as theta = 2 arccos((r - h)/r). For example, a tank with radius 1 meter, length 3 meters, and liquid height 0.5 meters: theta = 2 arccos(0.5) = 120 degrees = 2.094 radians, segment area = (1/2)(2.094 - 0.8660) = 0.6138 m^2, volume = 3 times 0.6138 = 1.841 m^3. This calculation is critical for inventory management in fuel storage, chemical processing, and water treatment facilities where tanks are often oriented horizontally.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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