Circle Segment Area Calculator
Our free linear algebra calculator solves circle segment area problems. Get worked examples, visual aids, and downloadable results.
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The segment area equals the sector area minus the triangle area. Sector area = (1/2)r^2*theta. Triangle area = (1/2)r^2*sin(theta). Subtracting gives A = (r^2/2)(theta - sin(theta)), where theta is in radians.
Last reviewed: December 2025
Worked Examples
Example 1: Quarter-Circle Segment
Example 2: Semicircular Segment (180 degrees)
Background & Theory
The Circle 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 Circle 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))
The segment area equals the sector area minus the triangle area. Sector area = (1/2)r^2*theta. Triangle area = (1/2)r^2*sin(theta). Subtracting gives A = (r^2/2)(theta - sin(theta)), where theta is in radians.
Worked Examples
Example 1: Quarter-Circle Segment
Problem: Find the segment area of a circle with radius 10 and central angle 90 degrees.
Solution: Convert angle: theta = 90 * pi/180 = pi/2 = 1.5708 rad\nSector area = (1/2)(10^2)(pi/2) = 50 * 1.5708 = 78.54\nTriangle area = (1/2)(10^2)(sin(pi/2)) = 50 * 1 = 50\nSegment area = 78.54 - 50 = 28.54\nChord = 2*10*sin(pi/4) = 20*0.7071 = 14.14\nSagitta = 10*(1 - cos(pi/4)) = 10*(1-0.7071) = 2.929
Result: Segment Area: 28.54 | Chord: 14.14 | Arc: 15.71 | Sagitta: 2.93
Example 2: Semicircular Segment (180 degrees)
Problem: Find the segment area of a circle with radius 8 and central angle 180 degrees.
Solution: Convert angle: theta = 180 * pi/180 = pi = 3.1416 rad\nSector area = (1/2)(64)(pi) = 100.53\nTriangle area = (1/2)(64)(sin(pi)) = 32 * 0 = 0\nSegment area = 100.53 - 0 = 100.53\nThis equals the semicircle area = pi*r^2/2 = 100.53\nChord = 2*8*sin(pi/2) = 16 (diameter)\nSagitta = 8*(1 - cos(pi/2)) = 8*1 = 8 (radius)
Result: Segment Area: 100.53 (semicircle) | Chord: 16 | Arc: 25.13 | Sagitta: 8
Frequently Asked Questions
What is a circle segment and how is it different from a sector?
A circle segment (also called a circular segment) is the region between a chord and the arc it subtends. Think of cutting a circle with a straight line (chord) - the smaller piece cut off is a minor segment, and the larger remaining piece is a major segment. A sector, on the other hand, is the region between two radii and the arc between them - shaped like a pizza slice. The key difference is that a segment is bounded by a chord and an arc, while a sector is bounded by two radii and an arc. The segment area equals the sector area minus the triangle area formed by the two radii and the chord.
What is the formula for the area of a circle segment?
The area of a circular segment is calculated as: A = (r^2/2)(theta - sin(theta)), where r is the radius and theta is the central angle in radians. This formula comes from subtracting the triangle area from the sector area. The sector area is (1/2)r^2*theta and the triangle area (formed by the two radii and the chord) is (1/2)r^2*sin(theta). When theta is in degrees, first convert to radians by multiplying by pi/180. For a semicircle (theta = pi or 180 degrees), the segment area equals the sector area since the triangle area is zero. The formula works for both minor and major segments.
What is the sagitta (segment height) and how is it calculated?
The sagitta is the maximum height of a circular segment, measured as the perpendicular distance from the midpoint of the chord to the arc. It is calculated using the formula h = r(1 - cos(theta/2)), where r is the radius and theta is the central angle. An alternative formula using the chord length c is: h = r - sqrt(r^2 - (c/2)^2). The sagitta is useful in engineering for measuring the curvature of arched structures, calculating lens profiles in optics, and determining road crown heights. For small angles, the sagitta approximates to h = c^2/(8r), which is the basis of the sagitta formula widely used in surveying.
How do you find the segment area given only the chord length and radius?
When you know the chord length c and radius r, first find the central angle using the relationship c = 2r*sin(theta/2), which gives theta = 2*arcsin(c/(2r)). Then apply the standard segment area formula: A = (r^2/2)(theta - sin(theta)). For example, with r = 10 and c = 14.14: theta = 2*arcsin(14.14/20) = 2*arcsin(0.707) = 2*(pi/4) = pi/2 = 90 degrees. Then A = (100/2)(pi/2 - sin(pi/2)) = 50(1.5708 - 1) = 28.54. This reverse calculation is common in construction and manufacturing where the chord and radius are the directly measurable quantities.
What are real-world applications of circle segment calculations?
Circle segment calculations appear in numerous practical applications across engineering and science. In civil engineering, they determine the cross-sectional area of partially filled pipes and channels for flow rate calculations. In architecture, they compute window areas for arched openings and the material needed for curved facades. In agriculture, they calculate irrigation coverage from rotating sprinklers. In optics, segment geometry determines lens properties and mirror curvature. In manufacturing, they help design cams, gears, and rounded components. Even in everyday life, calculating how much pizza you get from a specific slice angle is a segment calculation.
What is the difference between a minor segment and a major segment?
A minor segment is the smaller region created when a chord divides a circle, corresponding to a central angle less than 180 degrees (pi radians). A major segment is the larger region, corresponding to a central angle greater than 180 degrees. Together, the minor and major segments make up the entire circle area. When the central angle equals exactly 180 degrees, the chord is a diameter, and both segments are equal semicircles. The formulas work the same for both types: just use the appropriate central angle. For the major segment, you can either use theta > pi directly or calculate it as circle area minus the minor segment area.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy