Circle Segment Area Calculator
Our free linear algebra calculator solves circle segment area problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Circle Segment Area Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: A = (r^2/2)(theta - sin(theta))
Worked example — Segment Area: 28.54 | Chord: 14.14 | Arc: 15.71 | Sagitta: 2.93
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 Sector area = (1/2)(10^2)(pi/2) = 50 * 1.5708 = 78.54 Triangle area = (1/2)(10^2)(sin(pi/2)) = 50 * 1 = 50 Segment area = 78.54 - 50 = 28.54 Chord = 2*10*sin(pi/4) = 20*0.7071 = 14.14 Sagitta = 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 Sector area = (1/2)(64)(pi) = 100.53 Triangle area = (1/2)(64)(sin(pi)) = 32 * 0 = 0 Segment area = 100.53 - 0 = 100.53 This equals the semicircle area = pi*r^2/2 = 100.53 Chord = 2*8*sin(pi/2) = 16 (diameter) Sagitta = 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.
How is the arc length of a segment calculated?
The arc length of a circular segment is calculated using the formula s = r * theta, where r is the radius and theta is the central angle in radians. This formula derives from the fact that the full circumference (2*pi*r) corresponds to a full angle of 2*pi radians, so the arc length is proportional to the central angle. For an angle in degrees, the formula becomes s = (theta/360) * 2*pi*r = pi*r*theta/180. The arc length plus the chord length gives the perimeter of the segment. Understanding arc length is essential in road design for calculating curve distances, in gear design for determining tooth spacing, and in robotics for planning circular motion paths.
How do you calculate the chord length from the central angle?
The chord length is calculated using the formula c = 2*r*sin(theta/2), where r is the radius and theta is the central angle in radians. This formula comes from basic trigonometry: if you draw the two radii to the endpoints of the chord, you form an isosceles triangle. Dropping a perpendicular from the center to the chord bisects both the chord and the central angle, creating a right triangle where half the chord equals r*sin(theta/2). For a 60-degree central angle, the chord equals the radius (equilateral triangle). For 90 degrees, the chord equals r*sqrt(2). For 120 degrees, the chord equals r*sqrt(3). The maximum chord length is the diameter (2r) at 180 degrees.
Can the segment area formula be used for elliptical segments?
The circular segment formula does not directly apply to ellipses, but it can be adapted. For an ellipse with semi-axes a and b, the area of an elliptical sector is (ab/2)*theta (analogous to the circular case), but the angle theta must be the eccentric anomaly, not the geometric angle. The elliptical segment area requires integration: A = integral of y dx between the chord endpoints, where y follows the ellipse equation. A practical approximation scales the circular result: multiply the circular segment area by (b/a) if the chord is perpendicular to the major axis. Exact elliptical segment areas involve incomplete elliptic integrals, making numerical methods necessary for precise calculations.
How accurate is Circle Segment Area Calculator for very small or very large angles?
Circle Segment Area Calculator uses standard IEEE 754 double-precision floating point arithmetic, providing approximately 15 significant digits of accuracy. For very small angles (less than 0.01 degrees), the segment area approaches zero and the formula remains accurate, though relative errors increase as the area approaches machine epsilon. For angles approaching 360 degrees, the segment approaches the full circle area and the formula handles this correctly. The most sensitive region is near 0 and 360 degrees where the sin(theta) term nearly cancels the theta term. For practical engineering applications, the calculator provides more than sufficient accuracy across all angles from 0 to 360 degrees.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Segment Area Calculator
Calculate segment area with inputs, formulas, and instant results.
🧮Area of a Circle Calculator
Calculate area of acircle with inputs, formulas, and instant results.
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.
🧮Area of Crescent Calculator
Calculate area of crescent with inputs, formulas, and instant results.
🧮Cross Sectional Area Calculator
Calculate cross sectional area with inputs, formulas, and instant results.
🧮Shoelace Formula Polygon Area Calculator
Calculate shoelace formula polygon area with inputs, formulas, and instant results.