Segment Area Calculator
Solve segment area problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
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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).
Last reviewed: December 2025
Worked Examples
Example 1: Segment Area with 90-Degree Central Angle
Example 2: Semicircular Segment (180 Degrees)
Background & Theory
The Segment Area Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Segment Area Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy