Sector Arc Length Calculator

The Sector Arc Length Calculator measures the curved distance along the arc of a circular sector, from a small slice to the complete circle. It accepts seven different starting measurements, so you can use whichever value you already have. Follow these steps to get a correct answer in under a minute.

1. Choose the input method. Pick Radius + Central Angle, Diameter + Central Angle, Full Circle Circumference + Central Angle, Sector Area + Radius, Sector Area + Central Angle, Chord Length + Central Angle, or Radius + Known Arc Length. The input fields, labels, and unit lists change to match the method you pick.
2. Enter your measurement. Type the value in the input box. For area methods, use the squared unit list (mm², cm², m², km², in², ft², yd²).
3. Enter the central angle. Type the angle in degrees or radians using the DEG/RAD toggle, or drag the angle slider from 0° to 360°. The live diagram and the result update as you move it. In the Sector Area + Radius and Radius + Known Arc Length methods, the angle is worked out from your values instead.
4. Set the start and end angles if you need them. The default sweep starts at 0° and runs counterclockwise. Change the start angle, end angle, or direction to measure a sector anywhere on the circle. The sweep, not the position, decides the arc length. Identical start and end angles mean a zero-length arc, unless you use the full-circle option for a complete revolution.
5. Choose input and output units. The calculator converts your input into a consistent internal unit, calculates, then converts the answer into your output unit. The number changes with the unit; the length does not.
6. Pick a precision. Choose 2, 4, 6, or 8 decimal places. The default is 4. The calculator always uses full-precision π internally and rounds only the displayed result.
7. Click Calculate Now. The main result is the sector arc length, with its classification: zero-angle sector, minor sector, semicircle, major sector, or complete circle. Below it you get the radius, diameter, complete circumference, complementary arc, chord length, sector area, sector perimeter, circular segment area, arc percentage, both angle forms, and the minor vs major sector comparison.
8. Explore the diagram. The live sector diagram draws the full circle, shades the selected sector in gold, and marks the centre O, the radius lines, the chord, and the endpoints. Use the toggles to show or hide the radius lines, the chord, the centre point, the angle marker, and the complementary sector.
9. Copy, print, or reset. Use Copy Result to place the answer on your clipboard, Print Summary for a simple report, or Reset to clear the panel. The preset buttons load verified examples instantly, and the quick-select buttons jump straight to 45°, 60°, 90°, 120°, 180°, 270°, or 360°.

Try an example. Radius 6 with a 60° angle gives an arc of 6.2831853072 units. Radius 8 with 90° gives 12.5663706144 units. Radius 5 with 270° gives a major-sector arc of 23.5619449019 units. Diameter 14 with 120° gives 14.6607657168 units. Sector area 50 with radius 10 gives an arc of 10.0000000000 units, and radius 4 with a known arc of 6 units gives an angle of 85.9436692696°.

Important tips. Keep the angle unit consistent: 60° and 1.0471975512 radians are the same angle, but 60 radians is not 60°. Arc length is the curved distance only; the chord is a different measurement and is always shorter, except for a zero-length arc. If you enter a diameter, the calculator halves it to find the radius first. Sector area is not arc length: one is measured in square units and the other in plain units. A zero or negative measurement is not a valid sector, so the calculator rejects it instead of showing a misleading answer.

A circle is the set of all points at a fixed distance, the radius, from a centre point O. A circular sector is the wedge-shaped region cut out of a circle by two radii and the arc between them, like a slice of a pie. The sector arc length is the curved distance along the outer edge of that wedge, from one radius endpoint to the other, following the curve.

Picture a slice of pizza. The two straight cut edges are the radii, the pointed end is the centre O, and the crust is the arc. The arc length is the length of the crust. The straight line across the wide end of the slice, joining the two crust endpoints, is the chord, and it is shorter than the crust.

Every sector has the same parts, scaled by its radius. The centre O is the fixed point. The radius r runs from O to any point on the circle. The central angle θ is the angle at O between the two radii that bound the sector, and it decides what fraction of the circle the sector covers. The arc is the curved edge. The chord is the straight line between the arc endpoints. The sector area is the region inside the two radii and the arc.

Two radii always divide a circle into two sectors: the smaller one, the minor sector, and the larger one, the major sector. Together they make the complete circle. Which one you want depends on your problem, and the calculator shows both, so you never have to guess.

Sector arc length is a one-dimensional measurement, a length, so it is reported in plain units such as cm, m, or inches. It is not an area, and it is not an angle. Keeping those three quantities separate is the first step to getting every sector calculation right.

The sector arc length comes straight from the circumference of a full circle. The circumference of a complete circle is:

A sector with central angle θ covers the fraction θ / 360 of a complete 360° rotation, so its arc is the same fraction of the circumference:

Simplifying gives the degree form of the formula:

When the angle is measured in radians, the formula is simpler still, because a radian is defined by the radius itself:

Each symbol in the formulas has a precise meaning:

Both forms give the same answer. Use L = πrθ / 180 when your angle is in degrees, and L = rθ when it is in radians. The calculator converts degrees to radians first and then uses L = rθ internally, which is why the two modes always agree.

Using the diameter instead of the radius gives a second form of the same formula. Since d = 2r, then r = d / 2, and substituting gives L = dθ / 2 in radians, or L = πdθ / 360 in degrees.

Radius plus central angle is the most direct way to find a sector arc length. The worked example below uses a radius of 6 units and a central angle of 60°.

1. Identify the radius. r = 6 units.
2. Identify the central angle. θ = 60°.
3. Convert degrees to radians. θ = 60 × π / 180 = π / 3 = 1.0471975512 radians.
4. Write the arc-length formula. L = rθ.
5. Substitute the values. L = 6 × π / 3.
6. Calculate. L = 6.2831853072 units.
7. Round as needed. To two decimal places, L = 6.28 units.

The same steps work for any radius and angle. A radius of 8 units with 90° gives L = 8 × π / 2 = 12.5663706144 units. A radius of 5 units with 270° gives L = 5 × 3π / 2 = 23.5619449019 units. A radius of 1 unit with 60° gives L = 1.0471975512 units, which is exactly π / 3, a useful check: for r = 1, the arc length equals the angle in radians.

Rounding rule. Round only at the end. If you replace π with 3.14 before multiplying, a radius of 6 with 60° gives 6.28 exactly by luck here, but for a radius of 8 with 90° it gives 12.56 instead of 12.5663706144, an error of 0.0063706144. That error grows with the radius, so keep full precision through the calculation and round only the final displayed result.

When you know the diameter instead of the radius, halve it first, then apply the same formula. The worked example below uses a diameter of 14 units and a central angle of 120°.

1. Identify the diameter. d = 14 units.
2. Find the radius. r = d / 2 = 14 / 2 = 7 units.
3. Identify the central angle. θ = 120° = 2π / 3 = 2.0943951024 radians.
4. Write the formula. L = rθ, or directly L = πdθ / 360.
5. Substitute. L = 7 × 2.0943951024, or L = π × 14 × 120 / 360.
6. Calculate. L = 14.6607657168 units.
7. Round as needed. To two decimal places, L = 14.66 units.

Both routes agree exactly, which is a good self-check. The radius route gives 7 × 2.0943951024 = 14.6607657168. The diameter route gives π × 14 × 120 / 360 = 14.6607657168. If the two disagree, the diameter was not halved correctly.

Another example: a diameter of 20 units with 180° gives r = 10 units, so L = π × 10 = 31.4159265359 units, the semicircle arc. A diameter of 10 units with 60° gives r = 5 units, so L = 5.2359877560 units. The most common error in this calculation is forgetting the division by two and using the diameter as if it were the radius, which doubles the answer. For d = 14, that mistake gives 29.3215314336 instead of 14.6607657168.

Measuring tip. Diameters are often easier to measure in real life than radii, because you can measure straight across the circle. That is why the calculator accepts diameter input directly: measure across, enter it, and the calculator halves it for you.

Degrees are the everyday unit for angles. A complete circle contains 360°, so a central angle in degrees tells you directly what fraction of the full circle your sector covers. An angle of 90° covers one quarter of the circle, 180° covers one half, and 270° covers three quarters.

The degree formula is L = πrθ / 180, where θ is in degrees. It comes from the fraction idea: θ / 360 of the circumference 2πr, simplified.

Worked example. Radius 8 units, angle 90°.

1. Identify the values. r = 8 units, θ = 90°.
2. Find the fraction of the circle. 90 / 360 = 1 / 4, a quarter circle.
3. Write the formula. L = πrθ / 180.
4. Substitute. L = π × 8 × 90 / 180 = π × 8 / 2.
5. Calculate. L = 12.5663706144 units.
6. Check against the circumference. The full circumference is 2π × 8 = 50.2654824574 units, and one quarter of it is 50.2654824574 / 4 = 12.5663706144 units. The two agree.

More degree examples with a radius of 5 units: 30° gives L = 2.6179938780 units, 45° gives L = 3.9269908170 units, 120° gives L = 10.4719755120 units, and 270° gives L = 23.5619449019 units. Notice that doubling the angle doubles the arc length for the same radius, because arc length is directly proportional to the central angle.

Common degree mistake. Putting a degree value straight into L = rθ without converting gives a wildly wrong answer. For r = 5 and 60°, the unconverted calculation gives 5 × 60 = 300, but the correct answer is 5.2359877560. The formula L = rθ needs radians, so either convert first or use the degree form L = πrθ / 180.

A radian is the angle you get when the arc length equals the radius. If you wrap a length of string equal to the radius around the circle, the angle it subtends at the centre is exactly 1 radian. A complete circle contains 2π radians, about 6.2831853072 radians, because the full circumference is 2πr and dividing by r leaves 2π.

The radian formula is L = rθ, the simplest arc-length formula there is. It has no π in it, because the π is already inside the radian measure of the angle.

Worked example. Radius 6 units, angle π / 3 radians (which is 60°).

1. Identify the values. r = 6 units, θ = π / 3 = 1.0471975512 radians.
2. Write the formula. L = rθ.
3. Substitute. L = 6 × 1.0471975512.
4. Calculate. L = 6.2831853072 units.
5. Check against degrees. π / 3 radians is 60°, and the degree formula gives L = π × 6 × 60 / 180 = 6.2831853072 units. The two agree exactly.

More radian examples: a radius of 4 units with θ = π / 2 radians gives L = 4 × 1.5707963268 = 6.2831853072 units. A radius of 10 units with θ = π radians gives L = 10 × 3.1415926536 = 31.4159265359 units, the semicircle arc. A radius of 5 units with θ = 1 radian gives L = 5 units, because the arc equals the radius when the angle is 1 radian.

Why radians are preferred in mathematics. Calculus formulas for arc length, and the formulas for chord, sector, and segment area on this page, all use radians. Degrees are convenient for everyday measurement, but radians make the mathematics clean, which is why the calculator converts every angle to radians before it calculates.

Sector arc length and sector area sound similar, but they measure different things. Arc length is a one-dimensional curved distance, measured in plain units. Sector area is a two-dimensional region, the wedge of the circle cut out by two radii and the arc, measured in square units.

The formulas look alike, which is exactly why they get confused:

The sector area is the arc length multiplied by half the radius: A = ½ × r × L. That relationship is a useful check. For a radius of 6 units and 60°, the arc is 6.2831853072 units, and the sector area is ½ × 6 × 6.2831853072 = 18.8495559215 square units.

Units are the giveaway. If your answer should be a length, the unit is cm, m, or inches. If it should be an area, the unit is cm², m², or square inches. An arc-length answer in square units, or a sector-area answer in plain units, is certainly wrong. The calculator keeps the two separate and labels the units clearly.

A full-circle check. At 360°, the sector is the whole circle, so the sector area must equal πr². For r = 6, π × 36 = 113.0973355292 square units. The arc at the same time is the full circumference, 37.6991118431 units. Both come from the same radius and angle, but they answer different questions.

The perimeter of a sector is the distance all the way around its boundary. A proper sector has three boundary pieces: the curved arc, the first radius, and the second radius. So the sector perimeter is the arc length plus twice the radius:

Here θ is in radians. The two straight sides are each exactly one radius long, so they add 2r to the curved arc length.

Worked example. Radius 6 units, angle 60°. The arc is L = 6.2831853072 units. The two radii add 2 × 6 = 12 units. The sector perimeter is P = 6.2831853072 + 12 = 18.2831853072 units.

More examples. Radius 8 with 90°: arc 12.5663706144, perimeter 12.5663706144 + 16 = 28.5663706144 units. Radius 5 with 270°: arc 23.5619449019, perimeter 23.5619449019 + 10 = 33.5619449019 units. Radius 7 with 120°: arc 14.6607657168, perimeter 14.6607657168 + 14 = 28.6607657168 units.

Two special cases. For a complete circle, 360°, there are no radius sides in the boundary at all: the perimeter is simply the circumference, P = 2πr. For r = 6, that is 37.6991118431 units, not 37.6991118431 + 12. For a zero-angle sector, 0°, the geometry is degenerate: the two radii coincide and the arc has zero length, so do not present an ordinary sector perimeter for it. The calculator treats the zero-angle case as degenerate and does not report a normal perimeter value.

Why the distinction matters. If you are ordering trim for a sector-shaped panel, you need the perimeter, because the trim runs along the arc and both straight edges. If you are measuring only the curved edge, you need the arc length. Mixing the two up is one of the most common sector errors, and the calculator shows both values side by side so the difference is always visible.

Two radii divide a circle into two sectors. The smaller one is the minor sector, and the larger one is the major sector. They share the same two radii, and together they make the complete circle.

The central angle decides which is which. A central angle below 180° gives a minor sector. A central angle above 180° gives a major sector, and its angle is a reflex angle, larger than a straight angle. Exactly 180° gives a semicircle, which is neither minor nor major: both sectors are equal.

Minor and major are complements. For a smaller central angle θ, the minor arc is Lminor = rθ and the major arc is Lmajor = r(2π - θ). Their sum is always the complete circumference, and their sector areas add to the complete circle area:

Worked example. Radius 6 units, smaller angle 60°. The minor arc is L = 6 × π / 3 = 6.2831853072 units, and the minor sector area is 18.8495559215 square units. The major arc uses the reflex angle 360° - 60° = 300°, so L = 6 × 5π / 3 = 31.4159265359 units, and the major sector area is 94.2477796077 square units. The arcs add to 6.2831853072 + 31.4159265359 = 37.6991118431 units, the full circumference, and the areas add to 18.8495559215 + 94.2477796077 = 113.0973355292 square units, the full circle area. The minor sector is 16.67% of the circle and the major sector is 83.33%.

Another example. Radius 5 units, angle 270°. This is already the major sector: L = 23.5619449019 units and the sector area is 58.9048622548 square units. Its complementary minor sector uses 90°: L = 7.8539816340 units and the area is 19.6349540849 square units. The sum is 31.4159265359 units, the full circumference 2π × 5.

Do not confuse the two angles. The most common minor/major error is using the smaller angle when the problem asks for the major sector, or the other way round. Always check whether the angle you were given is the smaller one or the reflex one, and remember that the calculator shows both sectors side by side so you can pick the one your problem needs.

A sector and a circular segment are two different regions of a circle, and they are easy to confuse because both involve an arc. A sector is enclosed by two radii and an arc. A circular segment is enclosed by a chord and an arc. The segment is the part of the sector that lies beyond the chord, so it is smaller than the sector, except in the special cases below.

The formulas show the difference clearly:

The segment area is the sector area minus the triangular area between the two radii and the chord. Here θ is the central sweep in radians, between 0 and 2π. The formula handles both minor and major segments correctly: for a minor arc it gives the small cap, and for a major arc it gives the large region that covers most of the circle.

Worked examples. For r = 6 units and 60°, the sector area is 18.8495559215 square units and the segment area is ½ × 36 × (1.0471975512 - 0.8660254038) = 3.2610986534 square units. The segment is much smaller than the sector, because the triangle takes up most of the wedge. For r = 8 and 90°, the segment is 18.2654824574 square units. For r = 5 and 90°, the segment is 7.1349540849 square units.

Special cases. At 180°, sin θ = 0, so the segment area equals the sector area: ½ πr². For r = 5, both are 39.2699081699 square units, which is exactly half the circle. At 270°, sin θ = -1, so the segment is larger than the sector: for r = 5, the sector is 58.9048622548 square units and the segment is 71.4048622548 square units. At 360°, the chord vanishes and the segment is the whole circle, πr².

Segment vs sector, in one line. The sector is the wedge from the centre. The segment is the cap beyond the chord. If your problem involves the centre, you want the sector. If it involves only the arc and its chord, you want the segment.

Chord length and sector arc length both connect the same two points on a circle, but they take different paths. The arc follows the curve. The chord cuts straight across. The arc is almost always longer.

The formulas show the difference clearly:

The chord depends on the sine of half the angle, which grows more slowly than the angle itself, so the gap between arc and chord widens as the angle grows.

Three facts from the table are worth remembering. At 30°, the arc is only about 1.15% longer than the chord, so for small angles the two are close. At 180°, the chord is the diameter, 10 units, while the arc is πr = 15.7079632679 units. At 270°, the chord shrinks back to 7.0710678119 units, the same as at 90°, because the chord depends on the smaller angle between the endpoints, while the arc keeps growing with the sweep.

Special case. At 60°, the chord equals the radius exactly: c = 2r sin(30°) = 2r × 0.5 = r. For r = 6, the chord is 6 units, and the radius, chord, and the two radii form an equilateral triangle. This is a handy check and a favourite of exam setters.

Why it matters. If you are cutting a curved pipe, you need the arc length. If you are measuring the straight gap between two points on a circular part, you need the chord. Mixing them up is one of the most common circle errors, and the calculator shows both values together so the difference is always visible.

Sometimes neither the radius nor the angle is given. The calculator works backwards from four other measurements by deriving the missing value first, then applying L = rθ as usual. All seven methods give the same radius and the same arc length for the same sector, so you can use whichever measurement you actually have.

From full circle circumference. The full circumference is C = 2πr, so r = C / (2π). For C = 37.6991118431 units, r = 37.6991118431 / 6.2831853072 = 6 units. Then a 60° sector arc is L = 6 × π / 3 = 6.2831853072 units. Notice the shortcut: a 60° arc is one sixth of the circumference, and 37.6991118431 / 6 = 6.2831853072 units.

From chord length. The chord formula is c = 2r sin(θ / 2), so r = c / [2 sin(θ / 2)]. For a chord of 6 units with a 60° angle, r = 6 / (2 × sin 30°) = 6 / (2 × 0.5) = 6 units. Then L = 6.2831853072 units. Chord input only works when the angle is strictly between 0° and 360°: at 0° or 360° the chord is zero and no radius can be recovered, so the calculator rejects those angles for this method.

From sector area and radius. The sector area is A = ½r²θ, so θ = 2A / r². For a sector area of 50 square units with a radius of 10 units, θ = 2 × 50 / 100 = 1 radian = 57.2957795131°. Then L = 10 × 1 = 10.0000000000 units. This method needs the sector area to be no larger than the full circle area πr²: for r = 10, that limit is 314.1592653590 square units, so an area above that is rejected.

From sector area and angle. Rearranged the same way, r = √(2A / θ) with θ in radians. For a sector area of 18.8495559215 square units with a 60° angle, θ = 1.0471975512 radians, so r = √(2 × 18.8495559215 / 1.0471975512) = √(36) = 6 units. Then L = 6.2831853072 units. This method needs the angle as well as the area, because the same sector area can come from many radius and angle combinations.

From radius and known arc length. The angle is θ = L / r. For a radius of 4 units and a known arc of 6 units, θ = 6 / 4 = 1.5 radians = 85.9436692696°. Then the sector area is A = ½ × 16 × 1.5 = 12.0000000000 square units, and the perimeter is P = 6 + 8 = 14.0000000000 units. This method needs the arc to be no longer than the complete circumference 2πr: for r = 4, that limit is 25.1327412287 units.

Which input to use. Use whichever measurement you actually have. If two different measurements give different radii, one of the measurements is wrong, and that disagreement is worth investigating before trusting either answer.

Sector arc length is not an abstract classroom exercise. Anywhere a curved distance on a circle has to be measured, cut, built, or travelled, the same formula applies.

Mechanical engineering. Pipe bending starts with arc length: a pipe bent through 90° with a bend radius of 8 units needs 12.5663706144 units of pipe for the bend alone. Sheet metal rolled into a curved panel, a gear tooth profile, and a cam shape all need the curved length before the material is cut. Getting it wrong wastes material or leaves a part short.

Architecture and construction. Arched doorways, circular windows, domes, and curved staircases are dimensioned by their arcs. A semicircular arch with a radius of 10 units needs 31.4159265359 units of trim along the curve. A sector-shaped canopy with a radius of 6 units and a 60° angle needs 6.2831853072 units of edging along the curve and 18.2831853072 units around the whole boundary.

Roads and tracks. A circular road curve, a running track bend, or a railway curve is laid out by radius and central angle. The length of the curve decides the amount of asphalt, the length of rail, and the distance a vehicle travels through the bend. A 120° curve with a radius of 7 units is 14.6607657168 units long.

Pizza and food portioning. A pizza slice is a sector. If a pizza has a radius of 6 units, one slice cut at 60° has a crust length of 6.2831853072 units and an area of 18.8495559215 square units. Six such slices make the whole pizza, and their crusts add to 37.6991118431 units, the full circumference. Cutting equal slices is a sector division problem: equal angles give equal arcs and equal areas.

Clocks, wheels, and robotics. A clock hand sweeps an arc as time passes. A wheel rolling along the ground covers a distance equal to the arc it turns through. A robot arm rotating about a joint moves its gripper along a circular arc, and the distance it travels is rθ, which is why radians are the natural unit in robotics.

Computer graphics and CAD. Curves in CAD models, fonts, and animations are built from circular arcs. The length of an arc decides how a tool path is divided, how a curve is rendered, and how far an object moves along a circular path in an animation. A sector-shaped part in CAD is defined by exactly the radius and angle this calculator uses.

A note on real shapes. Not every curved structure is a perfect circle. Real arches can be elliptical or parabolic, and real roads use clothoid transitions. The sector formulas on this page apply exactly when the curve is a true circular arc, and they are a good approximation only when the curve is close to circular. Always confirm the shape before applying the formula.

Most wrong answers in sector arc length come from a small list of errors. Checking for each one takes seconds and saves most mistakes.

1. Using degrees directly in the radian formula. L = rθ needs radians. Putting 60° straight in gives 6 × 60 = 360 instead of 6.2831853072. Convert first, or use L = πrθ / 180.
2. Confusing radius with diameter. The formula needs r, not d. Using d = 14 as the radius doubles the answer: 29.3215314336 instead of 14.6607657168 for a 120° sector. Halve the diameter first.
3. Confusing sector area with arc length. Sector area is a two-dimensional region in square units. Arc length is a one-dimensional distance in plain units. For r = 6 and 60°, the sector area is 18.8495559215 square units and the arc is 6.2831853072 units. Different quantities, different units.
4. Confusing sector perimeter with arc length. The perimeter adds two radius sides: P = L + 2r. For r = 6 and 60°, the arc is 6.2831853072 units but the perimeter is 18.2831853072 units. Using the perimeter when the problem asks for the arc overstates the answer by 12 units.
5. Using the wrong central angle. A sector problem may give the angle between the chord and a tangent, or an inscribed angle, instead of the central angle. An inscribed angle is half the central angle for the same arc. If a problem gives an inscribed angle φ, the central angle is θ = 2φ, and the arc is L = r × 2φ. Using φ directly halves the answer.
6. Confusing minor and major sectors. An angle of 270° gives a major-sector arc of 23.5619449019 units for r = 5. Using the smaller angle 90° instead gives 7.8539816340 units, the minor arc. Check which sector the problem asks for.
7. Using an incorrect chord formula. The chord is c = 2r sin(θ / 2), with half the angle inside the sine. Using sin(θ) instead gives the wrong value: for r = 6 and 60°, the correct chord is 6 units, while 2 × 6 × sin(60°) = 10.3923048454 units.
8. Mixing linear and squared measurement units. A radius in cm and an angle in degrees is fine, but the answer comes out in cm. A sector area in cm² must be converted with the squared factor: 1 m² is 10,000 cm², not 100 cm². Mixing cm and inches in one calculation, or forgetting to convert the output unit, gives a number with the wrong size. Convert first, calculate second.
9. Rounding π too early. Replacing π with 3.14 before multiplying introduces an error that grows with the radius. Keep full-precision π through the calculation and round only the final displayed result.
10. Forgetting to convert the angle into radians. Every sector formula that uses θ without a 180 in it, including the sector area A = ½r²θ, the chord c = 2r sin(θ / 2), and the segment area A = ½r²(θ - sin θ), needs θ in radians. Degrees work only in the degree forms of the formulas.

A final check that catches most of these. Verify the result against the circumference. The arc must be between 0 and 2πr, it must be longer than the chord, the sector area must be between 0 and πr², and the minor plus major arcs must add to exactly 2πr. If any of those checks fails, one of the ten mistakes above is the cause.

These six examples cover the calculator's main input methods and sector types. Each one is calculated by the same engine that powers the tool, and each result has been verified independently.

Example 1: 60-degree sector with radius 6. Given r = 6 units, θ = 60° = π / 3 radians. Arc length L = 6 × π / 3 = 6.2831853072 units. Diameter d = 12 units. Chord c = 2 × 6 × sin(30°) = 6 units. Sector area A = ½ × 36 × π / 3 = 18.8495559215 square units. Sector perimeter P = 6.2831853072 + 12 = 18.2831853072 units. Segment area is 3.2610986534 square units. Complete circumference C = 37.6991118431 units.

Example 2: Quarter-circle sector with radius 8 and 90°. Given r = 8 units, θ = 90° = π / 2 radians. Arc length L = 8 × π / 2 = 12.5663706144 units. Chord c = 2 × 8 × sin(45°) = 11.3137084990 units. Sector area A = ½ × 64 × π / 2 = 50.2654824574 square units. Sector perimeter P = 12.5663706144 + 16 = 28.5663706144 units. Segment area is 18.2654824574 square units. The arc is exactly one quarter of the circumference 50.2654824574 units.

Example 3: Major sector with radius 5 and 270°. Given r = 5 units, θ = 270° = 3π / 2 radians. Arc length L = 5 × 3π / 2 = 23.5619449019 units. Chord c = 2 × 5 × sin(135°) = 7.0710678119 units. Sector area A = ½ × 25 × 3π / 2 = 58.9048622548 square units. Sector perimeter P = 23.5619449019 + 10 = 33.5619449019 units. Complementary minor arc, using 90°, is 7.8539816340 units, and the complementary sector area is 19.6349540849 square units. The two arcs add to 31.4159265359 units, the full circumference.

Example 4: Sector from diameter 14 and 120°. Given d = 14 units, θ = 120° = 2π / 3 radians. Radius r = 14 / 2 = 7 units. Arc length L = 7 × 2π / 3 = 14.6607657168 units. Chord c = 2 × 7 × sin(60°) = 12.1243556530 units. Sector area A = ½ × 49 × 2π / 3 = 51.3126800086 square units. Sector perimeter P = 14.6607657168 + 14 = 28.6607657168 units. Segment area is 30.0950576159 square units.

Example 5: Sector from area 50 and radius 10. Given sector area A = 50 square units, radius r = 10 units. Central angle θ = 2 × 50 / 100 = 1 radian = 57.2957795131°. Arc length L = 10 × 1 = 10.0000000000 units. Chord c = 2 × 10 × sin(0.5) = 9.5885107721 units. Sector perimeter P = 10.0000000000 + 20 = 30.0000000000 units. Segment area is 7.9264507596 square units. Complementary arc is 52.8318530718 units.

Example 6: Sector from radius 4 and known arc length 6. Given r = 4 units, L = 6 units. Central angle θ = 6 / 4 = 1.5 radians = 85.9436692696°. Sector area A = ½ × 16 × 1.5 = 12.0000000000 square units. Chord c = 2 × 4 × sin(0.75) = 5.4531100802 units. Sector perimeter P = 6 + 8 = 14.0000000000 units. Segment area is 4.0200401072 square units. Complementary arc is 19.1327412287 units, and the complementary sector area is 38.2654824574 square units.

Click any preset button in the calculator to load one of these examples and see the full step-by-step solution, the live diagram, and all the secondary results for that input.