Segment Area
Calculate the area of a circular segment using radius and central angle. Free online geometry tool using the formula A = 0.5 x r^2 x (alpha - sin(alpha)) for chord-based circle sections.
About This Calculator
The segment area calculator computes the area of a circular segment -- the region between a chord and its corresponding arc -- using the formula A = 0.5 x r^2 x (alpha - sin(alpha)), where r is the circle radius and alpha is the central angle in radians. The calculator accepts the angle in degrees and handles the conversion internally.
A segment differs from a sector: while a sector is a pizza-slice shape between two radii, a segment is the area below the chord. The formula subtracts the isosceles triangle area from the sector area: A = A_sector - A_triangle = (0.5 x r^2 x alpha) - (0.5 x r^2 x sin(alpha)). The tool also visualizes the segment as a proportion of the full circle using a pie chart.
Enter the radius in any unit and the central angle in degrees (between 0 deg and 360 deg). The central angle must be positive. Use this calculator for engineering problems like partially filled pipes, architectural arch designs, and geometry homework.
Frequently Asked Questions
What is a circular segment?
A circular segment is the region between a chord and the arc it subtends on a circle. It looks like a slice of a circle cut by a straight line. When the central angle is less than 180 deg, it is called a minor segment; when greater than 180 deg, it is a major segment.
What is the formula for circular segment area?
The segment area formula is A = 0.5 x r^2 x (alpha - sin(alpha)), where r is the radius and alpha is the central angle in radians. To convert from degrees to radians, multiply by pi/180. For example, with radius 10 units and angle 60 deg, the segment area is 0.5 x 100 x (pi/3 - sin(60 deg)) ≈ 9.06 square units.
What is the difference between a segment and a sector?
A sector is the region between two radii and the arc (like a pizza slice). A segment is the region between a chord and the arc. The segment is the sector minus the isosceles triangle formed by the two radii. The formula A = 0.5 x r^2 x (alpha - sin(alpha)) reflects this difference.
Can I find the chord length from the segment area?
Yes. Given the radius r and central angle alpha (in radians), the chord length is c = 2r x sin(alpha/2). The chord is the straight line that forms the base of the segment.
What is the arc length of a circular segment?
The arc length of a circular segment is L = r x alpha, where alpha is the central angle in radians. For example, a segment with radius 5 units and angle 90 deg (pi/2 radians) has an arc length of 5 x pi/2 ≈ 7.85 units.
Where is segment area used in real life?
Segment area calculations are used in engineering (cross-section of partially filled pipes and tanks), construction (archways, tunnel cross-sections), astronomy (lunar phases), and design (circular cutouts, decorative arches). It is especially useful for calculating the volume of liquid in a horizontal cylindrical tank.