Sist oppdatert: 19. august 2026
Segment Area Kalkulator
Skapere
Dharmendra SinghAnmeldere
Skapere
Dharmendra SinghAnmeldere
Quick Answer
The segment area calculator finds the area of a minor circular segment using A = (θ/360) × πr² − (1/2)r²sin(θ). It also shows the matching sector area, triangle area, chord length, and arc length so you can tell the difference between a cap-shaped segment and a full sector.
Use the segment area calculator to subtract the center triangle from the matching sector and get the area between the chord and the arc.
Viktige Punkter
- Segment area is the matching sector area minus the center triangle area.
- This page is scoped to minor segments with central angles up to 180°.
- Chord length and arc length help confirm that the geometry setup is sensible.
- Sector area is always larger than the corresponding minor segment area.
- A hand estimate plus the calculator output is a reliable error-catching workflow.
Skapere
Dharmendra SinghAnmeldere
Skapere
Dharmendra SinghAnmeldere
Formel
A(segment) = (θ / 360) × πr² − (1/2)r²sin(θ)
Hvor:
- A=Segment area(square units)
- θ=Central angle of the minor segment(degrees)
- r=Circle radius(units)
Løste eksempler
Radius 10 and central angle 60°
Find the minor segment cut from a circle of radius 10 by a 60° arc.
- 1Compute the matching sector area: (60/360) × π × 10² = 52.359878.
- 2Compute the triangle area: (1/2) × 10² × sin(60°) = 43.301270.
- 3Subtract triangle area from sector area: 52.359878 − 43.301270 = 9.058607.
Radius 8 and central angle 90°
A quarter-sector minus its triangle gives a right-angle segment.
- 1Sector area = (90/360) × π × 8² = 16π ≈ 50.265482.
- 2Triangle area = (1/2) × 8² × sin(90°) = 32.
- 3Segment area = 50.265482 − 32 = 18.265482.
Radius 12 and central angle 120°
A wider minor segment shows how quickly area grows with angle.
- 1Sector area = (120/360) × π × 12² = 48π ≈ 150.796447.
- 2Triangle area = (1/2) × 12² × sin(120°) ≈ 62.353829.
- 3Segment area = 150.796447 − 62.353829 = 88.442618.
Introduksjon
The Segment Area Calculator finds the area of the minor circular segment: the region trapped between a chord and the arc above it. That makes it different from a sector, which includes the two radii that reach the center. This page keeps that distinction explicit, shows the supporting sector and triangle pieces, and gives worked examples you can check by hand. It is useful for geometry classes, CAD sketches, lens-shaped cutouts, roadway curves, and any circle problem where you need the cap-shaped region rather than the whole slice.
What a circular segment is
A circular segment is the cap-shaped part of a circle bounded by a chord and the arc above that chord. If you draw the two radii to the arc endpoints, you get a sector. Remove the center triangle from that sector and the leftover region is the segment.
A segment is bounded by one chord and one arc.
A sector is bounded by two radii and one arc.
This calculator uses the minor segment only.
The central angle controls how much of the circle is cut off.
Why the segment formula works
The segment area comes from subtraction. First find the area of the sector with the same radius and central angle. Then find the area of the isosceles triangle formed by the two radii and the chord. The difference between those two regions is exactly the segment area.
Sector area uses the fraction θ/360 of the full circle.
Triangle area uses (1/2)r²sin(θ).
Subtracting removes the center triangle.
Keeping θ in degrees matches the page input.
Segment area versus sector area
Users often mix segment and sector problems because the diagrams look similar. A sector includes the center point and both radii, so its area is always larger than the matching minor segment area. If your teacher or drawing labels the region between the chord and arc only, this calculator is the right one; if the center is included, use the sector area calculator.
Sector area is the larger “slice.”
Segment area is the smaller “cap.”
Both use the same radius and central angle.
The triangle area is the bridge between them.
Worked example with radius 10 and angle 60°
For radius 10 and angle 60°, the sector area is one sixth of the full circle area 100π, which is about 52.359878. The triangle area is 50sin(60°), or about 43.301270. Subtracting gives a segment area of about 9.058607 square units. Because the chord is 10 units long, this is a neat example to verify by hand.
Find the sector first.
Find the triangle second.
Subtract in the final step.
Check the chord and arc as geometry sanity checks.
Common mistakes
The most common mistake is stopping after finding sector area. Another is typing a major-segment angle larger than 180° into a formula intended for the minor segment. Users also mix radians and degrees, or forget that the triangle formula needs the sine of the included angle, not the half-angle.
Do not confuse segment area with sector area.
Do not use a major angle in this minor-segment tool.
Do not switch to radians unless every formula part matches.
Do not round the triangle area too early.
Real-world uses for segment area
Circular segments appear in dome panels, bridge arches, pipe cross-sections that are partly filled, camera lens profiles, and roadway sight-distance diagrams. Engineers and designers often need the cap-shaped area because it represents material removed, fluid depth, or the exposed cross-section above a straight cut.
Partially filled pipes and tanks.
Lens and window cutouts.
Arches and curved framing details.
Geometry homework and exam practice.
Manual method versus calculator use
Solving one or two examples by hand is the best way to learn the structure: sector minus triangle. After that, a calculator helps when the radius or angle is messy, or when you also want the chord and arc immediately. A good workflow is estimate, calculate, and then compare the result with the matching sector area.
Estimate whether the segment is much smaller than the sector.
Use full precision until the final display step.
Check whether the angle really describes the minor segment.
Reuse the triangle and sector outputs as a self-check.
Hurtigreferansekort
Segment area quick reference
Hurtigreferanse • Segment Area Kalkulator
A(segment) = (θ/360) × πr² − (1/2)r²sin(θ)Gyldig rekkevidde: Use radius > 0 and a minor central angle 0° < θ ≤ 180°.
Vanlige Verdier
⚠ Pass på
- •Do not use this calculator for major segments larger than 180°.
- •Do not stop after computing sector area; subtract the triangle too.
- •Keep the radius and all reported lengths in the same unit system.
- •Avoid rounding intermediate values before the final subtraction.
Proff-tips
- →Compare the segment area to the sector area to check scale.
- →Use chord length to picture how wide the cut really is.
- →If θ = 90°, the triangle area simplifies because sin(90°) = 1.
- →Use the sector area calculator when the region includes the center point.
Ofte Stilte Spørsmål
What is the formula for the area of a circular segment?
For a minor segment with central angle θ in degrees, the area is sector area minus triangle area: A = (θ/360)πr² − (1/2)r²sin(θ).
How is segment area different from sector area?
Sector area includes the two radii and the center point. Segment area is only the cap between the chord and the arc, so it is the sector minus the center triangle.
Can I use an angle larger than 180°?
Not in this calculator. This page is intentionally scoped to the minor segment, so the angle must be greater than 0° and at most 180°.
Why does the calculator show triangle area too?
The triangle area is the piece removed from the sector to create the segment. Seeing it helps you check that the subtraction step was set up correctly.
Do I need to convert degrees to radians myself?
No. The input is in degrees, and the calculator converts internally before using the sine function and arc-length formula.
What if I only know the chord and height?
This page does not handle that alternate input method. In that case, use a chord- or sagitta-focused calculator and then convert to the needed angle or radius.
When should I use the sector area calculator instead?
Use the sector tool whenever the region includes the center of the circle and both radii, such as a pie-slice region rather than a cap-shaped segment.