Segment Area Calculator

Q: What is a circular segment?

A circular segment is the region of a circle enclosed between a chord (a straight line connecting two points on the circle) and the arc subtended by that chord. It looks like a slice cut from a circle with a straight knife. The central angle theta determines the size of the segment. When theta = 180 degrees, the segment becomes a semicircle.

Q: What is the formula for the area of a circular segment?

Area = (r squared / 2) times (theta minus sin theta), where r is the radius and theta is the central angle in radians. Example: r = 10, theta = 60 degrees = pi/3 radians. Area = (100/2) times (1.0472 minus sin(60 degrees)) = 50 times (1.0472 minus 0.8660) = 50 times 0.1812 = 9.059 sq units.

Q: How do I calculate the chord length of a circular segment?

Chord = 2 times r times sin(theta/2), where theta is the central angle in radians (or half of theta in degrees). Example: r = 10, theta = 60 degrees. Chord = 2 times 10 times sin(30 degrees) = 20 times 0.5 = 10 units. The chord is the straight line connecting the two endpoints of the arc.

Q: What is the sagitta (segment height) of a circular segment?

The sagitta is the perpendicular distance from the midpoint of the chord to the arc. Formula: h = r times (1 minus cos(theta/2)). Example: r = 10, theta = 60 degrees. h = 10 times (1 minus cos(30 degrees)) = 10 times (1 minus 0.8660) = 10 times 0.134 = 1.340 units. It is the maximum depth of the segment.

Q: What is the arc length of a circular segment?

Arc length = r times theta, where theta is the central angle in radians. Example: r = 8, theta = 1.0472 radians (60 degrees). Arc = 8 times 1.0472 = 8.378 units. The arc is the curved portion of the segment boundary.

Q: What is the difference between a circular segment and a circular sector?

A circular sector is the pie-slice region bounded by two radii and an arc (like a pizza slice). A circular segment is the region between a chord and the arc. The sector always includes the centre of the circle; the segment does not (unless the angle is 360 degrees). Sector area = (r squared times theta) / 2. Segment area = Sector area minus Triangle area = (r squared / 2)(theta minus sin theta).

Q: How do I convert degrees to radians for the segment formula?

Radians = degrees times pi / 180. Example: 60 degrees = 60 times 3.14159 / 180 = 1.0472 radians. 90 degrees = pi/2 = 1.5708 radians. 180 degrees = pi = 3.14159 radians. This calculator accepts degrees directly - it handles the conversion automatically.

Q: What is a minor segment versus a major segment?

A minor segment has a central angle less than 180 degrees (less than a semicircle). A major segment has a central angle greater than 180 degrees (more than a semicircle). Together a minor and major segment of the same chord make up the full circle. This calculator works for any angle from 0 to 360 degrees (exclusive).

Q: What happens to the segment area as the angle approaches 360 degrees?

As theta approaches 360 degrees (2pi radians), the segment area approaches the full circle area pi r squared. The formula Area = (r squared / 2)(theta minus sin theta) gives (r squared / 2)(2pi minus 0) = pi r squared when theta = 2pi. At theta = 180 degrees (pi radians), area = (r squared / 2)(pi minus 0) = pi r squared / 2, the semicircle area.

Q: Can the segment formula give a negative area?

No. The formula (r squared / 2)(theta minus sin theta) is always non-negative for 0 less than theta less than 2pi. This is because theta is always greater than sin(theta) for positive theta (except at theta = 0 where both are 0). The area is 0 only when theta = 0 (degenerate case with no segment).

Calculate area, chord, arc length, and sagitta of any circular segment from radius and angle.

What is a Circular Segment?

A circular segment is the region of a circle that lies between a chord and the arc that the chord subtends. It is formed when a straight line (the chord) cuts across a circle, dividing it into two parts. The smaller region is called the minor segment and the larger region (on the same side as the centre) is the major segment.

Unlike a circular sector (which is a pie-slice shape including the centre), a segment does not include the centre of the circle. The segment is bounded by exactly two curves: a straight line (the chord) and a curved arc. The size of the segment is determined by the central angle (theta), which is the angle at the centre of the circle subtended by the same chord.

The area of a circular segment is derived by subtracting the triangle formed by the two radii and the chord from the circular sector. Sector area = (r squared times theta) / 2. Triangle area = (r squared times sin theta) / 2. Therefore, segment area = (r squared / 2) times (theta minus sin theta). This elegant formula requires the angle in radians.

Beyond the area, this calculator also computes the chord length (the straight line between the two arc endpoints), the arc length (the curved boundary of the segment), and the sagitta (from the Latin for arrow) which is the height of the segment measured perpendicularly from the midpoint of the chord to the arc. The sagitta is particularly important in engineering for calculating the rise of curved structures like arched bridges and vault ceilings.

Circular segments appear in many practical contexts: the cross-sectional flow area of a partially filled pipe, the shape of a lens (the intersection of two circles), the submerged area of a cylindrical float, the cut area of a circular saw blade below the surface, and various architectural arches.

Formula and Derivation

Let r be the radius and theta be the central angle in radians.

Segment Area:

A = (r² / 2)(θ − sin θ)

r = Radius of the circle

θ = Central angle in radians (degrees × π / 180)

Derived as: Sector area − Triangle area = (r²θ/2) − (r² sinθ/2)

Chord Length:

Chord = 2r × sin(θ / 2)

Arc Length:

Arc = r × θ

Sagitta (Segment Height):

h = r × (1 − cos(θ / 2))

The perpendicular distance from the midpoint of the chord to the arc

Validation: 0 < θ < 2π radians (0° < θ < 360°)

How to Use This Calculator

Choose your angle unit - select “Degrees” (most common) or “Radians” depending on how your angle is expressed. The calculator handles the conversion internally.
Enter the radius - type the radius of the full circle from which the segment is cut. Use any consistent length unit.
Enter the central angle - type the angle at the centre. In degrees this must be between 0 and 360. In radians, between 0 and 2pi (approximately 6.2832).
Click Calculate - results appear showing segment area, chord length, arc length, and sagitta.
Use the formula note - this confirms both the degree and radian values used, plus the computed area formula, so you can verify the maths.

Example Calculations

Example 1 - Radius 10, Angle 60 degrees

Find all properties of a circular segment with radius 10 cm and central angle 60 degrees.

theta = 60° = pi/3 = 1.0472 rad. r = 10

Area = (100/2)(1.0472 − sin 60°) = 50 × (1.0472 − 0.8660) = 50 × 0.1812 = 9.059 cm²

Chord = 2 × 10 × sin(30°) = 20 × 0.5 = 10.000 cm

Arc = 10 × 1.0472 = 10.472 cm | Sagitta = 10 × (1 − cos 30°) = 10 × 0.134 = 1.340 cm

Area = 9.059 cm² | Chord = 10.000 cm | Arc = 10.472 cm | Sagitta = 1.340 cm

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Example 2 - Radius 5, Angle 120 degrees

A segment with radius 5 m and central angle 120 degrees.

theta = 120° = 2pi/3 = 2.0944 rad. r = 5

Area = (25/2)(2.0944 − sin 120°) = 12.5 × (2.0944 − 0.8660) = 12.5 × 1.2284 = 15.355 m²

Chord = 2 × 5 × sin(60°) = 10 × 0.8660 = 8.660 m

Arc = 5 × 2.0944 = 10.472 m | Sagitta = 5 × (1 − cos 60°) = 5 × 0.5 = 2.5 m

Area = 15.355 m² | Chord = 8.660 m | Arc = 10.472 m | Sagitta = 2.5 m

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Example 3 - Radius 8, Angle 1.0472 radians (60 degrees)

Same angle as Example 1 but entered in radians, showing the radian mode.

theta = 1.0472 rad (= 60°). r = 8

Area = (64/2)(1.0472 − sin(1.0472)) = 32 × (1.0472 − 0.8660) = 32 × 0.1812 = 5.799 sq units

Chord = 2 × 8 × sin(0.5236) = 16 × 0.5 = 8.000 units

Arc = 8 × 1.0472 = 8.378 units | Sagitta = 8 × (1 − cos 0.5236) = 8 × 0.134 = 1.072 units

Area = 5.799 sq units | Chord = 8.000 | Arc = 8.378 | Sagitta = 1.072

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Example 4 - Radius 6, Angle 90 degrees (quarter circle segment)

A 90-degree segment from a circle of radius 6 cm.

theta = 90° = pi/2 = 1.5708 rad. r = 6

Area = (36/2)(1.5708 − sin 90°) = 18 × (1.5708 − 1.0) = 18 × 0.5708 = 10.274 cm²

Chord = 2 × 6 × sin(45°) = 12 × 0.7071 = 8.485 cm

Arc = 6 × 1.5708 = 9.425 cm | Sagitta = 6 × (1 − cos 45°) = 6 × 0.2929 = 1.757 cm

Area = 10.274 cm² | Chord = 8.485 cm | Arc = 9.425 cm | Sagitta = 1.757 cm

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Frequently Asked Questions

What is a circular segment?+

A circular segment is the region of a circle enclosed between a chord (a straight line connecting two points on the circle) and the arc subtended by that chord. It looks like a slice cut from a circle with a straight knife. The central angle theta determines the size of the segment. When theta = 180 degrees, the segment becomes a semicircle.

What is the formula for the area of a circular segment?+

Area = (r squared / 2) times (theta minus sin theta), where r is the radius and theta is the central angle in radians. Example: r = 10, theta = 60 degrees = pi/3 radians. Area = (100/2) times (1.0472 minus sin(60 degrees)) = 50 times (1.0472 minus 0.8660) = 50 times 0.1812 = 9.059 sq units.

How do I calculate the chord length of a circular segment?+

Chord = 2 times r times sin(theta/2), where theta is the central angle in radians (or half of theta in degrees). Example: r = 10, theta = 60 degrees. Chord = 2 times 10 times sin(30 degrees) = 20 times 0.5 = 10 units. The chord is the straight line connecting the two endpoints of the arc.

What is the sagitta (segment height) of a circular segment?+

The sagitta is the perpendicular distance from the midpoint of the chord to the arc. Formula: h = r times (1 minus cos(theta/2)). Example: r = 10, theta = 60 degrees. h = 10 times (1 minus cos(30 degrees)) = 10 times (1 minus 0.8660) = 10 times 0.134 = 1.340 units. It is the maximum depth of the segment.

What is the arc length of a circular segment?+

Arc length = r times theta, where theta is the central angle in radians. Example: r = 8, theta = 1.0472 radians (60 degrees). Arc = 8 times 1.0472 = 8.378 units. The arc is the curved portion of the segment boundary.

What is the difference between a circular segment and a circular sector?+

A circular sector is the pie-slice region bounded by two radii and an arc (like a pizza slice). A circular segment is the region between a chord and the arc. The sector always includes the centre of the circle; the segment does not (unless the angle is 360 degrees). Sector area = (r squared times theta) / 2. Segment area = Sector area minus Triangle area = (r squared / 2)(theta minus sin theta).

How do I convert degrees to radians for the segment formula?+

Radians = degrees times pi / 180. Example: 60 degrees = 60 times 3.14159 / 180 = 1.0472 radians. 90 degrees = pi/2 = 1.5708 radians. 180 degrees = pi = 3.14159 radians. This calculator accepts degrees directly - it handles the conversion automatically.

What is a minor segment versus a major segment?+

A minor segment has a central angle less than 180 degrees (less than a semicircle). A major segment has a central angle greater than 180 degrees (more than a semicircle). Together a minor and major segment of the same chord make up the full circle. This calculator works for any angle from 0 to 360 degrees (exclusive).

What happens to the segment area as the angle approaches 360 degrees?+

As theta approaches 360 degrees (2pi radians), the segment area approaches the full circle area pi r squared. The formula Area = (r squared / 2)(theta minus sin theta) gives (r squared / 2)(2pi minus 0) = pi r squared when theta = 2pi. At theta = 180 degrees (pi radians), area = (r squared / 2)(pi minus 0) = pi r squared / 2, the semicircle area.

Can the segment formula give a negative area?+

No. The formula (r squared / 2)(theta minus sin theta) is always non-negative for 0 less than theta less than 2pi. This is because theta is always greater than sin(theta) for positive theta (except at theta = 0 where both are 0). The area is 0 only when theta = 0 (degenerate case with no segment).

🔗 Related Calculators

📌 Quick Tips

💡A circular segment is the region between a chord and the arc it cuts. For a semicircle (angle = 180 degrees), the segment area is exactly pi r squared / 2.

💡The sagitta (segment height) h = r(1 - cos(theta/2)) is the maximum depth of the segment measured from the chord to the arc.

💡For small angles, the segment area is approximately (r squared / 2)(theta - theta) for very small theta, growing as theta cubed / 6 times r squared.