Segment Area Calculator
Solve segment area problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
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\nSector area = (1/2)(10^2)(1.5708) = 78.5398\nTriangle area = (1/2)(10^2)(sin 90) = 50.0000\nSegment area = 78.5398 - 50.0000 = 28.5398\nChord length = 2(10)sin(45) = 14.1421\nArc 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\nSector area = (1/2)(25)(pi) = 39.2699\nTriangle area = (1/2)(25)(sin 180) = 0\nSegment area = 39.2699 - 0 = 39.2699\nThis equals pi*r^2/2 = 39.2699 (semicircle)\nChord = 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.