Toolarithm

Math Calculator

Circle Calculator

Start with the circle measurement you know.

Circle inputs

Choose the measurement you already know.

Each mode retains its own value, pi convention and source unit.

Source, angle and output choices

Enter a nonnegative finite decimal, scientific e source or integer fraction. Enter only the coefficient when the pi option is selected.

This selector reinterprets the source number; it does not rescale the source field.

Converts radius, diameter, circumference and any selected arc.

Converts disk and sector coverage independently, including international acres and hectares.

The angle covers at most one complete turn. Switching preserves each independent source.

From zero through three hundred sixty degrees.

Load a worked example

Examples restore their complete source record, unit choices and sector settings.

Exact expressions stay unchanged; tiny nonzero or very large values use at least six significant scientific digits.

Source strings support sixty significant decimal digits or sixty digits per fraction integer, one hundred eighty characters and scientific exponents from negative one hundred through positive one hundred. Exact coefficients have an eighteen-hundred-digit limit. Pi and radical expressions stay exact; certified pi bounds resolve decimal rounding. Complete arbitrary radical factorization is not promised. The diagram uses numerical screen positions.

Exact circle area

A=25π cm2A=25\pi\,\mathrm{cm}^{2}

Rounded area view

A≈78.539816 cm2A\approx78.539816\,\mathrm{cm}^{2}

Radius

5 cm5\,\mathrm{cm}

Diameter

10 cm10\,\mathrm{cm}

Circumference

10π cm10\pi\,\mathrm{cm}

Selected arc

52π cm\frac{5}{2}\pi\,\mathrm{cm}

Exact pi powers and verified roots are preserved. Each rounded value is independently checked; it is not reused as an intermediate input.

Interpreted known measurement

r=5 cmr=5\,\mathrm{cm}

The source unit reinterprets its numeric value. Output selectors convert lengths and areas independently; use compatible units when comparing formulas.

Circle and selected sector

Circle radius, diameter and selected sectorA normalized circle uses equal horizontal and vertical scales. A dashed diameter passes through the center, with a distinct radius guide. Gold shading and a dark outline identify the selected counterclockwise sector. A full turn uses the whole circle; zero or unresolved tiny angles have no invented sector. A zero circle is one point.Equal horizontal and vertical scale

The selected sector is measured counterclockwise from the positive horizontal radius, including reflex angles. Screen positions use numerical trigonometry; exact formulas determine the quantities.

Radius reference: r=5 cmr=5\,\mathrm{cm}.

The dashed guide marks a diameter; the dark gold guide marks one radius. The selected curved arc excludes the straight chord and the radial edges.

Circle calculation step by step

  1. Interpret the known measurement

    r=5 cm=120 mr=5\,\mathrm{cm}=\frac{1}{20}\,\mathrm{m}

    The numeric field is an ordinary value or an explicit coefficient of pi as selected. Length and area sources have different unit dimensions; no hidden source buffer is used.

  2. Recover the nonnegative radius

    r=rknown=(120 m)  ⟹  r=5 cmr=r_{\mathrm{known}}=\left(\frac{1}{20}\,\mathrm{m}\right)\;\Longrightarrow\;r=5\,\mathrm{cm}

    The selected source unit is converted exactly before the radius is expressed in the chosen output length unit. A radius is a length; area units use a squared length scale. Negative sources are rejected.

  3. Calculate the whole circle

    d=2r=10 cm,C=2πr=10π cm,A=πr2=25π cm2d=2r=10\,\mathrm{cm},\quad C=2\pi r=10\pi\,\mathrm{cm},\quad A=\pi r^2=25\pi\,\mathrm{cm}^{2}

    The exact radius, diameter, circumference and disk area retain rational, radical and pi factors. Their rounded views are separate certified displays, not inputs to the next formula.

  4. Use the selected central angle

    θ=12π rad,L=rθ=52π cm,Asector=12r2θ=254π cm2\theta=\frac{1}{2}\pi\,\mathrm{rad},\quad L=r\theta=\frac{5}{2}\pi\,\mathrm{cm},\quad A_{\mathrm{sector}}=\frac12r^2\theta=\frac{25}{4}\pi\,\mathrm{cm}^{2}

    The angle lies between zero and one complete turn. Degrees and pi-radians preserve exact pi factors. Reflex sectors, zero angles and exact full turns retain their stated meaning; an arc is not a chord or a complete sector perimeter.

  5. Verify the source and circle identity

    C2−4πA=0,Δsource=0C^2-4\pi A=0,\qquad \Delta_{\mathrm{source}}=0

    Exact algebra verifies the identity in compatible units and recovers the known source. Unit conversion uses exact rational ratios. Decimal precision changes presentation without changing either verification.

Inputs saved with this calculation

Known measurement
Radius
Active source record
Value: 5; format: Ordinary number; unit: Centimeters
Include sector
true
Angle convention
Degrees
Active angle source
90
Output length unit
Centimeters
Output area unit
Square centimeters
Rounded decimal places
6
Show diagram
true
Show conversions
true

All circle result details

Compare the interpreted source, exact whole-circle and sector quantities, radius square and zero verification residuals. Lengths and areas use independent output units.

Circle calculation details
QuantityExact valueRounded valueInterpretation
Known radius5 cm5\,\mathrm{cm}5 cm5\,\mathrm{cm}The active numeric source, explicit pi convention and source unit are preserved.
Radius5 cm5\,\mathrm{cm}5 cm5\,\mathrm{cm}The nonnegative distance from the center to the circle.
Diameter10 cm10\,\mathrm{cm}10 cm10\,\mathrm{cm}Twice the radius, crossing the center between opposite boundary points.
Circumference10π cm10\pi\,\mathrm{cm}≈  31.415927 cm\approx\;31.415927\,\mathrm{cm}The complete curved boundary, counted once.
Circle area25π cm225\pi\,\mathrm{cm}^{2}≈  78.539816 cm2\approx\;78.539816\,\mathrm{cm}^{2}The disk coverage, expressed in the independent output area unit.
Central angle in radians12π rad\frac{1}{2}\pi\,\mathrm{rad}≈  1.570796 rad\approx\;1.570796\,\mathrm{rad}Arc and sector formulas use radians, including exact pi factors from degrees or pi-radians.
Fraction of a full turn14\frac{1}{4}0.250.25The selected angle divided by a complete turn; no angle is silently wrapped or clamped.
Selected arc length52π cm\frac{5}{2}\pi\,\mathrm{cm}≈  7.853982 cm\approx\;7.853982\,\mathrm{cm}The selected counterclockwise curved arc; the straight chord is not included.
Selected sector area254π cm2\frac{25}{4}\pi\,\mathrm{cm}^{2}≈  19.634954 cm2\approx\;19.634954\,\mathrm{cm}^{2}Coverage of the selected sector, including a reflex sector when the angle exceeds a half turn.
Radius squared25 cm225\,\mathrm{cm}^{2}25 cm225\,\mathrm{cm}^{2}This square uses the output length unit squared, which can differ from the selected output area unit.
Circle identity verification0000The exact identity between squared circumference and four times pi times area is checked in compatible units before display conversion.
Known-source recovery0000Substitution recovers the original nonnegative source, using its exact unit ratio and pi convention.

Circumference in all length units

Seventeen exact physical length definitions. Change the output length selector to convert every displayed length, including the radius and selected arc.

Exact circumference conversions
Length unitExact circumferenceRounded circumference
Nanometers108π nm10^{8}\pi\,\mathrm{nm}≈  314159265.358979 nm\approx\;314159265.358979\,\mathrm{nm}
Micrometers100000π μm100000\pi\,\mathrm{\mu m}≈  314159.265359 μm\approx\;314159.265359\,\mathrm{\mu m}
Millimeters100π mm100\pi\,\mathrm{mm}≈  314.159265 mm\approx\;314.159265\,\mathrm{mm}
Centimeters10π cm10\pi\,\mathrm{cm}≈  31.415927 cm\approx\;31.415927\,\mathrm{cm}
Decimetersπ dm\pi\,\mathrm{dm}≈  3.141593 dm\approx\;3.141593\,\mathrm{dm}
Meters110π m\frac{1}{10}\pi\,\mathrm{m}≈  0.314159 m\approx\;0.314159\,\mathrm{m}
Decameters1100π dam\frac{1}{100}\pi\,\mathrm{dam}≈  0.031416 dam\approx\;0.031416\,\mathrm{dam}
Hectometers11000π hm\frac{1}{1000}\pi\,\mathrm{hm}≈  0.003142 hm\approx\;0.003142\,\mathrm{hm}
Kilometers110000π km\frac{1}{10000}\pi\,\mathrm{km}≈  0.000314 km\approx\;0.000314\,\mathrm{km}
Mils (one thousandth inch)500000127π mil\frac{500000}{127}\pi\,\mathrm{mil}≈  12368.475014 mil\approx\;12368.475014\,\mathrm{mil}
Inches500127π in\frac{500}{127}\pi\,\mathrm{in}≈  12.368475 in\approx\;12.368475\,\mathrm{in}
International feet125381π ft\frac{125}{381}\pi\,\mathrm{ft}≈  1.030706 ft\approx\;1.030706\,\mathrm{ft}
International yards1251143π yd\frac{125}{1143}\pi\,\mathrm{yd}≈  0.343569 yd\approx\;0.343569\,\mathrm{yd}
International miles25402336π mi\frac{25}{402336}\pi\,\mathrm{mi}≈  0.000195 mi\approx\;0.000195\,\mathrm{mi}
International nautical miles118520π nmi\frac{1}{18520}\pi\,\mathrm{nmi}≈  0.00017 nmi\approx\;0.00017\,\mathrm{nmi}
Typographic points (one seventy-second inch)36000127π pt\frac{36000}{127}\pi\,\mathrm{pt}≈  890.530201 pt\approx\;890.530201\,\mathrm{pt}
Typographic picas (one sixth inch)3000127π pc\frac{3000}{127}\pi\,\mathrm{pc}≈  74.21085 pc\approx\;74.21085\,\mathrm{pc}

Circle area in all area units

Nineteen physical area definitions include international acres and hectares. The output area selector also converts the selected sector. Hidden conversion rows stay in CSV.

Exact circle area conversions
Area unitExact circle areaRounded circle area
Square nanometers25×1014π nm225\times10^{14}\pi\,\mathrm{nm}^{2}≈  7853981633974483.096157 nm2\approx\;7853981633974483.096157\,\mathrm{nm}^{2}
Square micrometers25×108π μm225\times10^{8}\pi\,\mathrm{\mu m}^{2}≈  7853981633.974483 μm2\approx\;7853981633.974483\,\mathrm{\mu m}^{2}
Square millimeters2500π mm22500\pi\,\mathrm{mm}^{2}≈  7853.981634 mm2\approx\;7853.981634\,\mathrm{mm}^{2}
Square centimeters25π cm225\pi\,\mathrm{cm}^{2}≈  78.539816 cm2\approx\;78.539816\,\mathrm{cm}^{2}
Square decimeters14π dm2\frac{1}{4}\pi\,\mathrm{dm}^{2}≈  0.785398 dm2\approx\;0.785398\,\mathrm{dm}^{2}
Square meters1400π m2\frac{1}{400}\pi\,\mathrm{m}^{2}≈  0.007854 m2\approx\;0.007854\,\mathrm{m}^{2}
Square decameters140000π dam2\frac{1}{40000}\pi\,\mathrm{dam}^{2}≈  0.000079 dam2\approx\;0.000079\,\mathrm{dam}^{2}
Square hectometers14×106π hm2\frac{1}{4\times10^{6}}\pi\,\mathrm{hm}^{2}≈  0.000001 hm2\approx\;0.000001\,\mathrm{hm}^{2}
Square kilometers14×108π km2\frac{1}{4\times10^{8}}\pi\,\mathrm{km}^{2}≈  7.853982×10−9 km2\approx\;7.853982\times10^{-9}\,\mathrm{km}^{2}
Square mils (one thousandth inch)625×10816129π mil2\frac{625\times10^{8}}{16129}\pi\,\mathrm{mil}^{2}≈  12173695.880052 mil2\approx\;12173695.880052\,\mathrm{mil}^{2}
Square inches6250016129π in2\frac{62500}{16129}\pi\,\mathrm{in}^{2}≈  12.173696 in2\approx\;12.173696\,\mathrm{in}^{2}
Square international feet15625580644π ft2\frac{15625}{580644}\pi\,\mathrm{ft}^{2}≈  0.08454 ft2\approx\;0.08454\,\mathrm{ft}^{2}
Square international yards156255225796π yd2\frac{15625}{5225796}\pi\,\mathrm{yd}^{2}≈  0.009393 yd2\approx\;0.009393\,\mathrm{yd}^{2}
Square international miles625647497027584π mi2\frac{625}{647497027584}\pi\,\mathrm{mi}^{2}≈  3.032439×10−9 mi2\approx\;3.032439\times10^{-9}\,\mathrm{mi}^{2}
Square international nautical miles11371961600π nmi2\frac{1}{1371961600}\pi\,\mathrm{nmi}^{2}≈  2.289855×10−9 nmi2\approx\;2.289855\times10^{-9}\,\mathrm{nmi}^{2}
Square typographic points (one seventy-second inch)324×10616129π pt2\frac{324\times10^{6}}{16129}\pi\,\mathrm{pt}^{2}≈  63108.439442 pt2\approx\;63108.439442\,\mathrm{pt}^{2}
Square typographic picas (one sixth inch)225000016129π pc2\frac{2250000}{16129}\pi\,\mathrm{pc}^{2}≈  438.253052 pc2\approx\;438.253052\,\mathrm{pc}^{2}
International acres31255058570528π acre\frac{3125}{5058570528}\pi\,\mathrm{acre}≈  0.000002 acre\approx\;0.000002\,\mathrm{acre}
Hectares14×106π ha\frac{1}{4\times10^{6}}\pi\,\mathrm{ha}≈  0.000001 ha\approx\;0.000001\,\mathrm{ha}
What Can You Create?

Connect a known measurement to the whole circle

Recover the circle from the information supplied

Choose radius, diameter, circumference or area rather than rearranging every formula outside the tool. Each known quantity has an explicit source unit and its own retained input record. Circumference and area also distinguish an ordinary number from a coefficient of pi. The interpreted source and radius-recovery steps show exactly which geometric problem is being solved, helping prevent a diameter, circumference or area from being reused as though it were a radius.

Preserve the exact mathematical relationship

An ordinary circumference can introduce reciprocal pi factors, and an ordinary area can introduce a root involving pi. Those expressions remain exact beside their numerical views. Fractional and scientific sources are preserved through unit ratios and formula rearrangement. The calculation checks recovery of the known source and the identity relating circumference and disk area before a displayed decimal is used for estimation or copied elsewhere.

Study a specified arc and sector

Choose the central angle convention explicitly, then compare the selected curved arc with its enclosed sector. Exact degrees and multiples of pi radians preserve their pi factors. Reflex sectors, zero angles and full turns have different graphical states, while unresolved tiny arcs are explained without inventing visible thickness. Disable sector calculations when only the whole circle is needed; inactive angle sources then stop affecting validation and sharing.

Formulas and worked examples

Circle formulas and step-by-step examples

Start with a radius and calculate the whole circle

A radius of 5 cm5\,\mathrm{cm} gives diameter 10 cm10\,\mathrm{cm}, circumference 10π cm10\pi\,\mathrm{cm} and disk area 25π cm225\pi\,\mathrm{cm}^2. The circumference is a curved length; the area is coverage. Their different dimensions explain why a length conversion and an area conversion cannot use the same unsquared factor.

d=2(5)=10 cm,C=2π(5)=10π cm,A=π(5)2=25π cm2d=2(5)=10\,\mathrm{cm},\quad C=2\pi(5)=10\pi\,\mathrm{cm},\quad A=\pi(5)^2=25\pi\,\mathrm{cm}^2

Distinguish an ordinary circumference from a pi coefficient

If C=10π cmC=10\pi\,\mathrm{cm}, the radius is exactly 5 cm5\,\mathrm{cm}. If C=10 cmC=10\,\mathrm{cm} instead, the radius is 5/π cm5/\pi\,\mathrm{cm} and area is 25/π cm225/\pi\,\mathrm{cm}^2. The coefficient selector changes the meaning of the source; it is not a cosmetic display option.

10π2π=5,102π=5π,π(5π)2=25π\frac{10\pi}{2\pi}=5,\qquad \frac{10}{2\pi}=\frac5\pi,\qquad \pi\left(\frac5\pi\right)^2=\frac{25}\pi

Recover a radius from a known area

For an ordinary area 2 cm22\,\mathrm{cm}^2, the radius is 2/π cm\sqrt{2/\pi}\,\mathrm{cm} and the circumference is 22π cm2\sqrt{2\pi}\,\mathrm{cm}. If the known area is 2π cm22\pi\,\mathrm{cm}^2, the radius is instead 2 cm\sqrt2\,\mathrm{cm}. The nonnegative root is selected in both cases.

r=Aπ,A=2⟹r=2π,A=2π⟹r=2r=\sqrt{\frac A\pi},\qquad A=2\Longrightarrow r=\sqrt{\frac2\pi},\qquad A=2\pi\Longrightarrow r=\sqrt2

Convert a quarter turn before using arc and sector formulas

The default angle 90∘90^\circ equals π/2\pi/2 radians and one quarter of a full turn. With radius 5 cm5\,\mathrm{cm}, the selected arc is 5π/2 cm5\pi/2\,\mathrm{cm} and sector area is 25π/4 cm225\pi/4\,\mathrm{cm}^2. A reflex angle 270∘270^\circ instead selects three quarters of the circle, while 360∘360^\circ selects the complete disk and curved boundary.

θ=90π180=π2,L=5π2=5π2 cm,Asector=12(5)2π2=25π4 cm2\theta=90\frac{\pi}{180}=\frac\pi2,\qquad L=5\frac\pi2=\frac{5\pi}{2}\,\mathrm{cm},\qquad A_{\mathrm{sector}}=\frac12(5)^2\frac\pi2=\frac{25\pi}{4}\,\mathrm{cm}^2

Convert compatible units and check the circle identity

The default radius converts to 1/20 m1/20\,\mathrm{m}, and its disk area becomes π/400 m2\pi/400\,\mathrm{m}^2. Exact recovery of a known source is checked separately from C2=4πAC^2=4\pi A. That identity must use compatible units; a circumference measured in centimeters cannot be squared and directly compared with an area left in square meters without conversion.

5 cm=120 m,25π cm2=π400 m2,(10π)2−4π(25π)=05\,\mathrm{cm}=\frac1{20}\,\mathrm{m},\qquad 25\pi\,\mathrm{cm}^2=\frac\pi{400}\,\mathrm{m}^2,\qquad (10\pi)^2-4\pi(25\pi)=0
Why Users Love This Tool

Why keep source meaning, units and exact results together?

The input keeps its geometric meaning

  • A radius is distinguished from a full diameter, and known area sources use area units rather than length labels.
  • An ordinary circumference or area is explicitly different from the same numeric coefficient multiplied by pi.
  • Angle conventions preserve separate sources, with one nonnegative turn enforced instead of automatic wrapping.
  • A zero circle is one point; a tiny sector or missing gap is not replaced by a misleading full or positive region.

The working remains inspectable

  • Rational coefficients, nonnegative roots and required pi powers survive source recovery and physical conversion.
  • Length and area outputs have independent selectors, with exact international and stated typographic definitions.
  • Certified interval endpoints must agree on rounded displays, while algebra separately verifies the source and circle identity.
  • Complete rows, ordered equations, source records, conversion tables, sharing, CSV and printing preserve the problem beyond a single rounded number.
Perfect For

Circle reasoning for learning and measured diagrams

Students learning inverse circle formulas

Start from the measurement actually given in a problem and follow the radius-recovery step. Compare an ordinary circumference with a pi-multiple circumference to see why apparently similar sources can define different circles. Switch between a known radius and known disk area without confusing their units. Optional sector working connects degrees to radians and shows why curved arc length, straight chord length and covered area answer different geometric questions.

Teachers building reproducible examples

Use independent source buffers and explicit angle conventions to prepare checked fractional, radical or pi-based examples. The native diagram has equal axis scales and marks a full diameter, one radius and the selected sector without claiming numerical screen positions are exact symbolic coordinates. Zero, reflex and full-turn cases make useful boundary examples. Share or print the active record and complete equations so students can inspect the assumptions as well as the final result.

People comparing idealized circular dimensions

Estimate the dimensions or coverage of a flat circular disk when one compatible measurement is known. Convert lengths and areas independently into a useful scale while retaining the source unit. Check whether a real measurement refers to an internal radius, outside diameter, curved edge or complete coverage. Wall thickness, annular holes, irregular boundaries, measurement uncertainty, purchasing increments and material waste require additional models rather than being silently added to an ideal circle.

How It Works

How it works in three quick steps.

1

Identify the measurement and its unit

Choose radius, diameter, circumference or area as the known measurement. Enter a nonnegative finite decimal, scientific source or integer fraction, and select an appropriate source unit. Circumference and area can be ordinary numbers or exact coefficients of pi; those are different defined inputs. Read the interpreted source before using the result. Each mode retains its own source record when you switch.

2

Recover the radius and inspect the exact circle

Follow the radius-recovery formula for the selected measurement. The tool preserves rational factors, roots and pi powers instead of rounding an intermediate radius. Inspect the diameter, circumference and disk area in independently chosen output length and area units. Exact algebra checks the circle identity in compatible units and recovers the original nonnegative source, providing a reproducible record of the calculation.

3

Add a defined sector and save the working

Include arc and sector calculations when needed. Select degrees, ordinary radians or multiples of pi radians, with an angle from zero through one full turn. Follow the radian-based formulas and read the diagram caption, especially for reflex, zero, full-turn or unresolved tiny sectors. Copy the summary, share active sources, download all circumference and area conversions as CSV or print the mathematical working to a PDF.

Download & Print

Save the circle definition with its exact working

Download the source and every conversion

CSV preserves the known measurement, pi convention, source and output units, optional active angle, exact quantities, rounded views, radius square, source-recovery residual and circle-identity residual. It also retains all seventeen circumference conversions and nineteen disk-area conversions when their visual tables are hidden. Long coefficients remain complete in the file. Keep the source record with an exported value so an ordinary number is not later mistaken for a coefficient of pi.

Print mathematical steps or save a PDF

The print layout removes editing controls while retaining the interpreted source, whole-circle quantities, available diagram, optional sector working, source record and complete result tables. Browser printing can save a PDF with repeated table headings and readable grouped rows. Select a practical paper orientation. Very long numerical expressions use an explicit compact notice pointing to CSV instead of being silently clipped or reduced to an unreadably small string.

Share one complete active problem

The shared fragment includes the selected known measurement and its value, meaning and unit, plus output units and active optional angle. Inactive measurement and angle buffers are excluded, and malformed, duplicate, hidden or extra imported fields are rejected. Local switching still preserves the complete editor buffers. Plain copying provides readable mathematics outside the visual renderer, along with method notes that distinguish exact expressions, rounded displays and numerical screen geometry.

FAQ

Frequently Asked Questions

Can I calculate a circle from circumference or area?
Yes. Choose any of the four known measurements: radius, diameter, circumference or area. The corresponding radius relationships are r=rknownr=r_{\mathrm{known}}, r=d/2r=d/2, r=C/(2π)r=C/(2\pi) and r=A/πr=\sqrt{A/\pi}. The nonnegative branch is selected because a geometric radius cannot be negative. Each workflow preserves its own numeric source, pi convention and unit. A value retained in an inactive workflow does not affect the selected circle.
What does coefficient of pi mean?
The coefficient option multiplies your numeric source by pi exactly. For example, circumference entered as coefficient 1010 means C=10πC=10\pi, whereas an ordinary source 1010 means C=10C=10. They give different radii, 55 and 5/π5/\pi. Type only the coefficient in the numeric field; do not type a formula there. The interpreted known measurement is typeset beside the result so the chosen meaning stays visible.
Why can an ordinary area produce a root involving pi?
If the source is ordinary area A=2A=2, recovering the radius gives r=2/πr=\sqrt{2/\pi} in the corresponding length scale. That is an exact expression, not a calculation failure. If the source instead means A=2πA=2\pi, the radius is 2\sqrt2. The tool retains the required root and pi power, with a separately rounded numerical view. Choosing decimal precision does not replace the source or alter the symbolic circle.
How do degrees, radians and pi-radians differ?
Degrees divide a full turn into 360360 parts, while a full turn is 2π2\pi radians. In the multiples-of-pi mode, entering 1/21/2 means π/2\pi/2 radians and matches 90∘90^\circ. In ordinary radians, entering 1/21/2 means only 0.50.5 radians. Independent angle buffers preserve those different sources. The calculator converts the active convention explicitly and rejects angles outside a single nonnegative turn rather than silently wrapping them.
What is the difference between arc length, chord and sector area?
The selected arc follows the circle's curved boundary and has length L=rθL=r\theta when the central angle is measured in radians. A chord is a straight segment connecting its endpoints and is a different length. The sector is the region between the two radii and selected arc, with area Asector=r2θ/2A_{\mathrm{sector}}=r^2\theta/2. Neither arc length nor sector area is a complete sector perimeter, which would require a separate boundary interpretation.
Does the calculator support reflex sectors and full turns?
Yes. A central angle greater than a half turn and less than one full turn describes the larger, reflex sector. The native diagram uses the corresponding large arc. An exact full turn covers the whole disk and uses a complete circle rather than a same-endpoint arc path. A zero angle gives zero selected arc and sector area. Very tiny positive arcs or missing gaps can be omitted visually while their exact quantities remain available.
Why are source area units and output length units separate?
A known area is a coverage measurement, while the recovered radius and circumference are lengths. The source selector therefore offers area units when Area is selected and length units otherwise. Output lengths and areas are independently converted through exact physical ratios. Seventeen length and nineteen area definitions include international customary units, stated typographic definitions, acres and hectares. Changing a source unit reinterprets its numeric value; changing an output unit converts the result.
Can I use fractions, scientific notation, zero or unitless values?
Sources accept nonnegative finite decimals, scientific e notation and integer fractions with nonzero denominators. They support sixty significant decimal digits or sixty digits per fraction integer, one hundred eighty characters and exponents from negative one hundred through positive one hundred. Zero gives a collapsed circle with zero lengths and coverage. Unitless sources require unitless outputs and receive no invented physical conversion. Negative, missing and malformed active sources remain invalid instead of becoming zero.
How are exact results and decimal rounding checked?
The engine keeps rational coefficients, verified nonnegative roots and symbolic half-integer pi powers. A certified six-hundred-place pi interval bounds the squared quantity, and integer square-root comparisons check the rounded view at both endpoints. Both views must agree or the precision limit is reported. Rational ties round halfway away from zero. Common root factors are extracted with bounded work, without promising complete factorization of arbitrary large integers. Exact source recovery and a circle identity are checked separately.
Can I share the calculation or save it as a PDF?
Share the selected known measurement, explicit pi convention, source and output units, active optional angle and display choices through a reproducible link. Inactive source buffers are excluded. Copy summary provides readable plain-text mathematics, and CSV retains complete working plus every circumference and disk-area conversion even when the tables are hidden. Browser printing can save a PDF with the available diagram and full tables. Very long printed numbers use an explicit notice directing you to complete CSV values.
About This Tool

About this exact circle and sector calculator

This circle calculator starts with the information supplied rather than assuming that every problem begins with a radius. Four independent workflows accept radius, diameter, circumference or disk area, with an explicit distinction between ordinary values and coefficients of pi where that distinction is useful. Recovering a nonnegative radius defines the complete idealized circle. The resulting diameter, circumference and coverage remain connected through exact algebra, and independent output selectors keep length and area units dimensionally clear. The interpreted source makes those assumptions visible before a numerical result is reused.

Exact expression handling matters when inverse formulas introduce reciprocal or square-root powers of pi. The engine preserves rational source strings, verified nonnegative roots and symbolic half-integer pi powers instead of rounding an intermediate radius and propagating that approximation into every later quantity. Certified rational bounds separately check the rounded numerical displays. Source recovery and the circle identity provide additional algebraic checks in compatible units. Common radical factors are extracted with bounded work, while the interface states input and coefficient limits rather than promising arbitrary symbolic simplification.

Optional sector calculations add a specified nonnegative angle from zero through a full turn. Degrees, ordinary radians and pi-radians remain distinct source conventions. The diagram identifies a counterclockwise selected sector, handles reflex and exact full-turn cases explicitly, and declines to invent visible thickness for an unresolved tiny arc or gap. A zero circle collapses to one point. Sharing, plain copying, complete conversion CSV and native printing preserve the active geometry and its interpretation. Real annular regions, curved surfaces, irregular shapes, physical tolerances and material-planning allowances require additional information beyond the ideal disk and selected sector modeled here.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

Equations are checked by substituting worked examples into the stated relationship and comparing the result with the interactive implementation.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

Keep building

Explore more calculators

Calculators hub