Sector Area Calculator With Arc, Chord, Segment, and Reverse Solving

Enter Sector Measurements

Choose a direct or reverse calculation. Fractions and scientific notation are accepted.

Calculation mode

Radius Distance from center to circle.

Diameter The radius equals half this value.

Inner radius

Outer radius

Central angle

Known sector area

Known arc length

Known sector perimeter

Sector percentage

Length unit

Area result unit

Sector interpretation

Decimal precision

Number format

Custom π value Use a value between three and four.

Display separator

Normalize angles
Reduce large angles to one revolution.

Remove trailing zeros
Keep result formatting compact.

Live preview
Refresh the diagram while typing.

Calculation Tools

Apply a common central angle instantly.

Add api=1 to the query string. The same file returns structured JSON.

Generated API URL

Formula Used

Sector area in degrees

A = (θ ÷ 360°) × πr²

The angle represents a fraction of the full circle.

Sector area in radians

A = ½r²θ

The angle must be measured in radians.

Arc length

L = rθ

Use radians for this compact formula.

Sector perimeter

P = 2r + L

Add the arc and both straight radii.

Chord length

c = 2r sin(θ ÷ 2)

The minor and major sectors share this chord.

Segment area

Asegment = ½r²(θ − sin θ)

This removes the central triangle from the sector.

Major sector area

Amajor = πr² − Aminor

Both complementary areas equal the full circle.

Annular sector area

A = ½(R² − r²)θ

Subtract the inner sector from the outer sector.

Reverse radius formula

r = √(2A ÷ θ)

This solves radius from known area and angle.

How to Use This Calculator

Select the calculation mode matching your known measurements.
Enter the radius, diameter, area, arc, or perimeter.
Choose the correct length, area, and angle units.
Select minor, major, both, or direct interpretation.
Set precision and your preferred number notation.
Press calculate and review the formulas and diagram.
Download, copy, print, share, or save the result.

Batch Sector Calculator

Paste rows using radius,angle,length unit,angle unit,sector type.

Batch input

#	Radius	Angle	Area	Arc	Perimeter	Status

Example Data Table

These examples use the standard value of π.

Radius	Angle	Sector area	Arc length
5 cm	90°	19.635 cm²	7.854 cm
10 cm	45°	39.270 cm²	7.854 cm
8 m	120°	67.021 m²	16.755 m
12 in	180°	226.195 in²	37.699 in
3 ft	270°	21.206 ft²	14.137 ft
7.5 yd	30°	14.726 yd²	3.927 yd

Sector Area Guide

Understanding Circle Sectors

A circle sector looks like a carefully cut pizza slice. Two radii form its straight boundaries. A curved arc closes the remaining edge. The central angle controls the sector's share. A larger angle creates a larger sector. Radius growth also increases area quickly. This calculator handles both influences accurately.

Using Degrees and Radians

Angles commonly appear in degrees or radians. A full circle contains 360 degrees. The same circle contains two pi radians. Degree calculations use a circle fraction. Radian calculations use one-half times radius squared. Both formulas produce identical results after conversion. Gradians and turns are converted internally too.

Minor and Major Sectors

A minor sector normally uses the shorter arc. Its central angle stays below 180 degrees. A major sector uses the longer arc. Its central angle exceeds 180 degrees. Both sectors share the same chord endpoints. Their combined areas equal the entire circle. Selecting both displays a complete comparison.

Area, Arc, and Perimeter

Sector area measures the enclosed surface. Arc length measures only the curved boundary. Sector perimeter includes that arc and two radii. These measurements use related angle fractions. The calculator reports them together for convenience. This helps verify drawings, patterns, and manufactured parts.

Circular Segments and Chords

A chord joins the arc endpoints directly. The radii and chord form a triangle. Removing that triangle from a sector creates a segment. Segment calculations require trigonometric functions. The angle is converted to radians first. Major segments remain supported without manual rearrangement.

Reverse Calculations

Real problems often provide area instead of radius. Others provide perimeter or arc length. Reverse modes solve these missing quantities directly. Each mode validates the supplied measurements. The displayed steps show every rearranged formula. This reduces mistakes during homework and design checks.

Annular Sector Applications

An annular sector resembles part of a washer. It uses separate inner and outer radii. Examples include curved paths and ring plates. Area equals the outer sector minus the inner sector. Arc lengths are reported for both boundaries. The radial side lengths are also included.

Practical Uses

Sector measurements appear in architecture and manufacturing. Designers use them for fans and signs. Surveyors use them for circular land regions. Students use them throughout geometry lessons. Garden layouts may contain curved planting beds. Accurate units remain important in every application.

Reliable Measurement Practices

Use one consistent length unit whenever possible. Confirm whether an angle is minor or major. Measure radius from the exact circle center. Avoid confusing diameter with radius. Keep extra decimals during intermediate calculations. Round only the final result. The calculator offers precision controls for this purpose.

Frequently Asked Questions

What is a sector of a circle?

A sector is the region between two radii and their connecting arc. It resembles a pizza slice. Its size depends on the radius and central angle.

Can this calculator use radians?

Yes. Select radians beside the angle field. Degrees, gradians, and complete turns are also supported. Values convert automatically during calculation.

How is a major sector different?

A major sector uses the longer arc. Its angle exceeds 180 degrees. The minor and major areas together equal the full circle area.

Can I calculate an unknown radius?

Yes. Choose a reverse-solving mode. Enter the known area, arc length, perimeter, or angle. The calculator rearranges the appropriate formula.

What is an annular sector?

An annular sector lies between two concentric circles. Enter inner and outer radii. The calculator subtracts the smaller sector from the larger sector.

Why does the chord stay unchanged?

Complementary minor and major sectors share identical endpoints. Therefore, both use the same straight chord, although their arcs and areas differ.

Can results be downloaded?

Yes. Download a CSV report or a generated PDF. You can also print, copy, share, and restore previous calculations from local history.

Does the calculator accept fractions?

Yes. Inputs like 3/4, 7/2, and 1.25 are accepted. Scientific notation, including 2.5e3, is also supported.

Extended Sector Reference Table

This reference offers ready-made checks for development and classroom use.

Radius	Angle	Area	Arc length
1	15°	0.13089969	0.26179939
1	30°	0.26179939	0.52359878
1	45°	0.39269908	0.78539816
1	60°	0.52359878	1.04719755
1	75°	0.65449847	1.30899694
1	90°	0.78539816	1.57079633
1	120°	1.04719755	2.09439510
1	135°	1.17809725	2.35619449
1	150°	1.30899694	2.61799388
1	180°	1.57079633	3.14159265
1	210°	1.83259571	3.66519143
1	240°	2.09439510	4.18879020
1	270°	2.35619449	4.71238898
1	300°	2.61799388	5.23598776
2	15°	0.52359878	0.52359878
2	30°	1.04719755	1.04719755
2	45°	1.57079633	1.57079633
2	60°	2.09439510	2.09439510
2	75°	2.61799388	2.61799388
2	90°	3.14159265	3.14159265
2	120°	4.18879020	4.18879020
2	135°	4.71238898	4.71238898
2	150°	5.23598776	5.23598776
2	180°	6.28318531	6.28318531
2	210°	7.33038286	7.33038286
2	240°	8.37758041	8.37758041
2	270°	9.42477796	9.42477796
2	300°	10.47197551	10.47197551
2.5	15°	0.81812309	0.65449847
2.5	30°	1.63624617	1.30899694
2.5	45°	2.45436926	1.96349541
2.5	60°	3.27249235	2.61799388
2.5	75°	4.09061543	3.27249235
2.5	90°	4.90873852	3.92699082
2.5	120°	6.54498469	5.23598776
2.5	135°	7.36310778	5.89048623
2.5	150°	8.18123087	6.54498469
2.5	180°	9.81747704	7.85398163
2.5	210°	11.45372322	9.16297857
2.5	240°	13.08996939	10.47197551
2.5	270°	14.72621556	11.78097245
2.5	300°	16.36246174	13.08996939
3	15°	1.17809725	0.78539816
3	30°	2.35619449	1.57079633
3	45°	3.53429174	2.35619449
3	60°	4.71238898	3.14159265
3	75°	5.89048623	3.92699082
3	90°	7.06858347	4.71238898
3	120°	9.42477796	6.28318531
3	135°	10.60287521	7.06858347
3	150°	11.78097245	7.85398163
3	180°	14.13716694	9.42477796
3	210°	16.49336143	10.99557429
3	240°	18.84955592	12.56637061
3	270°	21.20575041	14.13716694
3	300°	23.56194490	15.70796327
4	15°	2.09439510	1.04719755
4	30°	4.18879020	2.09439510
4	45°	6.28318531	3.14159265
4	60°	8.37758041	4.18879020
4	75°	10.47197551	5.23598776
4	90°	12.56637061	6.28318531
4	120°	16.75516082	8.37758041
4	135°	18.84955592	9.42477796
4	150°	20.94395102	10.47197551
4	180°	25.13274123	12.56637061
4	210°	29.32153143	14.66076572
4	240°	33.51032164	16.75516082
4	270°	37.69911184	18.84955592
4	300°	41.88790205	20.94395102
5	15°	3.27249235	1.30899694
5	30°	6.54498469	2.61799388
5	45°	9.81747704	3.92699082
5	60°	13.08996939	5.23598776
5	75°	16.36246174	6.54498469
5	90°	19.63495408	7.85398163
5	120°	26.17993878	10.47197551
5	135°	29.45243113	11.78097245
5	150°	32.72492347	13.08996939
5	180°	39.26990817	15.70796327
5	210°	45.81489286	18.32595715
5	240°	52.35987756	20.94395102
5	270°	58.90486225	23.56194490
5	300°	65.44984695	26.17993878
6	15°	4.71238898	1.57079633
6	30°	9.42477796	3.14159265
6	45°	14.13716694	4.71238898
6	60°	18.84955592	6.28318531
6	75°	23.56194490	7.85398163
6	90°	28.27433388	9.42477796
6	120°	37.69911184	12.56637061
6	135°	42.41150082	14.13716694
6	150°	47.12388980	15.70796327
6	180°	56.54866776	18.84955592
6	210°	65.97344573	21.99114858
6	240°	75.39822369	25.13274123
6	270°	84.82300165	28.27433388
6	300°	94.24777961	31.41592654
7.5	15°	7.36310778	1.96349541
7.5	30°	14.72621556	3.92699082
7.5	45°	22.08932335	5.89048623
7.5	60°	29.45243113	7.85398163
7.5	75°	36.81553891	9.81747704
7.5	90°	44.17864669	11.78097245
7.5	120°	58.90486225	15.70796327
7.5	135°	66.26797004	17.67145868
7.5	150°	73.63107782	19.63495408
7.5	180°	88.35729338	23.56194490
7.5	210°	103.08350895	27.48893572
7.5	240°	117.80972451	31.41592654
7.5	270°	132.53594007	35.34291735
7.5	300°	147.26215564	39.26990817
8	15°	8.37758041	2.09439510
8	30°	16.75516082	4.18879020
8	45°	25.13274123	6.28318531
8	60°	33.51032164	8.37758041
8	75°	41.88790205	10.47197551
8	90°	50.26548246	12.56637061
8	120°	67.02064328	16.75516082
8	135°	75.39822369	18.84955592
8	150°	83.77580410	20.94395102
8	180°	100.53096491	25.13274123
8	210°	117.28612573	29.32153143
8	240°	134.04128655	33.51032164
8	270°	150.79644737	37.69911184
8	300°	167.55160819	41.88790205
10	15°	13.08996939	2.61799388
10	30°	26.17993878	5.23598776
10	45°	39.26990817	7.85398163
10	60°	52.35987756	10.47197551
10	75°	65.44984695	13.08996939
10	90°	78.53981634	15.70796327
10	120°	104.71975512	20.94395102
10	135°	117.80972451	23.56194490
10	150°	130.89969390	26.17993878
10	180°	157.07963268	31.41592654
10	210°	183.25957146	36.65191429
10	240°	209.43951024	41.88790205
10	270°	235.61944902	47.12388980
10	300°	261.79938780	52.35987756
12	15°	18.84955592	3.14159265
12	30°	37.69911184	6.28318531
12	45°	56.54866776	9.42477796
12	60°	75.39822369	12.56637061
12	75°	94.24777961	15.70796327
12	90°	113.09733553	18.84955592
12	120°	150.79644737	25.13274123
12	135°	169.64600329	28.27433388
12	150°	188.49555922	31.41592654
12	180°	226.19467106	37.69911184
12	210°	263.89378290	43.98229715
12	240°	301.59289474	50.26548246
12	270°	339.29200659	56.54866776
12	300°	376.99111843	62.83185307
15	15°	29.45243113	3.92699082
15	30°	58.90486225	7.85398163
15	45°	88.35729338	11.78097245
15	60°	117.80972451	15.70796327
15	75°	147.26215564	19.63495408
15	90°	176.71458676	23.56194490
15	120°	235.61944902	31.41592654
15	135°	265.07188015	35.34291735
15	150°	294.52431127	39.26990817
15	180°	353.42917353	47.12388980
15	210°	412.33403578	54.97787144
15	240°	471.23889804	62.83185307
15	270°	530.14376029	70.68583471
15	300°	589.04862255	78.53981634
20	15°	52.35987756	5.23598776
20	30°	104.71975512	10.47197551
20	45°	157.07963268	15.70796327
20	60°	209.43951024	20.94395102
20	75°	261.79938780	26.17993878
20	90°	314.15926536	31.41592654
20	120°	418.87902048	41.88790205
20	135°	471.23889804	47.12388980
20	150°	523.59877560	52.35987756
20	180°	628.31853072	62.83185307
20	210°	733.03828584	73.30382858
20	240°	837.75804096	83.77580410
20	270°	942.47779608	94.24777961
20	300°	1047.19755120	104.71975512
25	15°	81.81230869	6.54498469
25	30°	163.62461737	13.08996939
25	45°	245.43692606	19.63495408
25	60°	327.24923475	26.17993878
25	75°	409.06154344	32.72492347
25	90°	490.87385212	39.26990817
25	120°	654.49846950	52.35987756
25	135°	736.31077819	58.90486225
25	150°	818.12308687	65.44984695
25	180°	981.74770425	78.53981634
25	210°	1145.37232162	91.62978573
25	240°	1308.99693900	104.71975512
25	270°	1472.62155637	117.80972451
25	300°	1636.24617374	130.89969390
30	15°	117.80972451	7.85398163
30	30°	235.61944902	15.70796327
30	45°	353.42917353	23.56194490
30	60°	471.23889804	31.41592654
30	75°	589.04862255	39.26990817
30	90°	706.85834706	47.12388980
30	120°	942.47779608	62.83185307
30	135°	1060.28752059	70.68583471
30	150°	1178.09724510	78.53981634
30	180°	1413.71669412	94.24777961
30	210°	1649.33614313	109.95574288
30	240°	1884.95559215	125.66370614
30	270°	2120.57504117	141.37166941
30	300°	2356.19449019	157.07963268
50	15°	327.24923475	13.08996939
50	30°	654.49846950	26.17993878
50	45°	981.74770425	39.26990817
50	60°	1308.99693900	52.35987756
50	75°	1636.24617374	65.44984695
50	90°	1963.49540849	78.53981634
50	120°	2617.99387799	104.71975512
50	135°	2945.24311274	117.80972451
50	150°	3272.49234749	130.89969390
50	180°	3926.99081699	157.07963268
50	210°	4581.48928649	183.25957146
50	240°	5235.98775598	209.43951024
50	270°	5890.48622548	235.61944902
50	300°	6544.98469498	261.79938780