Arc Length Calculator with Steps

Arc length is the distance along a curve between two points. For a circle, it is the length of the curved portion of the circumference cut off by two radii. Picture a pizza slice. The crust edge of that slice is an arc, and its length is the arc length. The two straight edges are radii, and the angle between them at the center is called the central angle.

The central angle determines what fraction of the full circle your arc represents. A full circle measures 360 degrees, or 2π radians. If your central angle is 90 degrees, your arc is one quarter of the circumference. If the angle is 180 degrees, the arc is half the circumference. This proportional relationship is the foundation of every arc length formula.

Arcs come in two sizes for any pair of points on a circle. The minor arc is the shorter path between the points, corresponding to a central angle less than 180 degrees. The major arc is the longer path going the other way around, corresponding to an angle greater than 180 degrees. Unless stated otherwise, arc length problems refer to the minor arc.

A simple everyday example helps. Suppose a circular garden has a radius of 10 meters, and you want to install edging along a section that spans a 90-degree angle from the center. The full circumference is 2π(10), about 62.83 meters. A 90-degree angle is one quarter of the full circle, so the edging needed is about 15.71 meters. That curved distance is the arc length.

The arc length formula depends on whether your angle is measured in radians or degrees. Both versions express the same idea: arc length is a fraction of the full circumference, and that fraction is determined by the central angle.

When the angle θ is in radians, the formula is simple:

s = rθ

Here s is the arc length, r is the radius, and θ is the central angle in radians. This formula works so cleanly because a radian is defined as the angle that cuts off an arc equal in length to the radius. An angle of 1 radian on a circle of radius 5 gives an arc of exactly 5 units. The units of s always match the units of r. If the radius is in centimeters, the arc length is in centimeters.

When the angle is in degrees, the formula needs a conversion factor:

s = πrθ / 180

Here θ is the central angle in degrees. The π/180 factor converts degrees to radians behind the scenes, because 180 degrees equals π radians. So dividing by 180 and multiplying by π turns your degree measure into the radian measure the formula actually needs.

Both formulas come from the circumference of a full circle, C = 2πr. A full circle corresponds to a central angle of 2π radians or 360 degrees. The arc length is the same fraction of the circumference as the central angle is of the full angle. In radians: s / (2πr) = θ / (2π), which simplifies to s = rθ. In degrees: s / (2πr) = θ / 360, which simplifies to s = πrθ / 180.

Converting between degrees and radians is straightforward. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. For example, 60 degrees becomes 60 × π/180 = π/3 radians, about 1.047 radians. An angle of 2 radians becomes 2 × 180/π, about 114.59 degrees.

The table below summarizes the key formulas for quick reference.

Example 1: Angle in Degrees

Find the arc length of a circle with radius 7 cm and a central angle of 60 degrees.

Step 1: Identify the given values. The radius r is 7 cm and the angle θ is 60 degrees.

Step 2: Choose the correct formula. Since the angle is in degrees, use s = πrθ / 180.

Step 3: Substitute the values. This gives s = π(7)(60) / 180.

Step 4: Simplify. The fraction 60/180 reduces to 1/3, so s = 7π/3 cm. This is the exact answer.

Step 5: Find the decimal approximation. Since π is about 3.14159, s is about 7(3.14159)/3, which equals about 7.33 cm.

The arc length is 7π/3 cm exactly, or approximately 7.33 cm.

Example 2: Angle in Radians

Find the arc length of a circle with radius 5 meters and a central angle of 1.2 radians.

Step 1: Identify the given values. The radius r is 5 m and the angle θ is 1.2 radians.

Step 2: Choose the correct formula. Since the angle is in radians, use s = rθ.

Step 3: Substitute the values. This gives s = 5(1.2).

Step 4: Multiply. The result is s = 6 meters.

The arc length is exactly 6 meters. Notice how much simpler the radian formula is. There is no π to deal with and no fraction to simplify. This is one reason mathematicians and scientists prefer radians for theoretical work.

Not every curve is part of a circle. For a general smooth curve described by a function y = f(x), the arc length between x = a and x = b is found with an ∫:

L = ∫[a to b] √(1 + [f'(x)]²) dx

Here f'(x) is the derivative of the function. The formula comes from a simple idea. Break the curve into many tiny straight segments. Each segment has length √((Δx)² + (Δy)²) by the Pythagorean theorem. Factor out Δx to get √(1 + (Δy/Δx)²) Δx. As the segments become infinitely small, Δy/Δx becomes the derivative f'(x), and the sum becomes the ∫ above.

The derivative matters because it measures how steep the curve is at each point. A steep section contributes more length than a flat section over the same horizontal distance. The √(1 + [f'(x)]²) factor is sometimes called the local stretch factor. When the curve is flat, f'(x) is zero and the factor is 1, meaning the arc length equals the horizontal distance. When the curve is steep, the factor is larger than 1.

The limits a and b mark where you start and stop measuring along the x-axis. It is important to get these in the right order, with a as the smaller value. Swapping them changes the sign of the result.

Many arc length integrals cannot be evaluated with elementary antiderivatives. In those cases, numerical methods like Simpson's rule or the trapezoidal rule give accurate approximations. The calculator on this page uses numerical integration when a symbolic answer is not available and clearly labels the result as approximate.

Worked Example: y = x^3/2 from x = 0 to x = 4

Find the arc length of the curve y = x^3/2 over the interval [0, 4].

Step 1: Find the derivative. Using the power rule, f'(x) = (3/2)x^1/2.

Step 2: Square the derivative. [f'(x)]² = (9/4)x.

Step 3: Substitute into the formula. L = ∫[0 to 4] √(1 + (9/4)x) dx.

Step 4: Evaluate the ∫. Let u = 1 + (9/4)x, so du = (9/4)dx. When x = 0, u = 1. When x = 4, u = 10. The ∫ becomes (4/9)∫[1 to 10] √u du = (4/9)(2/3)[u^3/2] from 1 to 10 = (8/27)(10^3/2 - 1).

Step 5: Simplify. Since 10^3/2 = 10√10 ≈ 31.62, L ≈ (8/27)(30.62) ≈ 9.07.

The exact arc length is (8/27)(10√10 - 1), approximately 9.07 units.

Arc length responds to changes in radius and angle in a predictable way. Because the formula s = rθ is a simple product, each variable has a direct proportional effect.

If you double the radius while keeping the angle fixed, the arc length doubles. A 60-degree arc on a circle of radius 7 cm measures about 7.33 cm. On a circle of radius 14 cm, the same 60-degree angle cuts off an arc of about 14.66 cm. Every point on the larger circle is twice as far from the center, so the curved path between the same two angles is twice as long.

If you double the central angle while keeping the radius fixed, the arc length also doubles. A 30-degree arc on a 10 cm circle is about 5.24 cm. A 60-degree arc on the same circle is about 10.47 cm. Twice the angle means twice the fraction of the circumference.

A complete revolution, 360 degrees or 2π radians, gives the full circumference. Plugging θ = 2π into s = rθ yields s = 2πr, exactly the circumference formula. This confirms the formula is consistent: at the maximum angle, the "arc" is the entire circle.

Use the sliders in the interactive diagram to explore these relationships. Drag the angle slider and watch the highlighted arc grow. Drag the radius slider and see the whole circle expand. The displayed arc length updates in real time, reinforcing the proportional relationships described above.

Arc length appears in many practical situations where curved distances matter.

Running tracks. A standard 400-meter running track has two straight sections and two semicircular ends. If each semicircular end has a radius of 36.5 meters, the curved portion of one end is half the circumference: π(36.5), about 114.67 meters. Both ends together contribute about 229.35 meters of curved running. Track designers use these calculations to ensure each lane measures the correct total distance.

Bicycle wheels. A bicycle wheel with a diameter of 70 cm has a radius of 35 cm. In one full revolution, a point on the tire travels the full circumference: 2π(35), about 219.91 cm, or about 2.20 meters. If the wheel rotates 100 times, the bicycle travels about 220 meters. This relationship between rotations and distance is how cycle computers estimate speed and distance from wheel sensors.

Architectural arches. A semicircular arch over a doorway has a radius of 1.5 meters. The curved length of the arch is half the circumference: π(1.5), about 4.71 meters. A builder ordering curved trim for this arch needs this measurement to buy the right amount of material.

Rotating machinery. A conveyor belt wraps halfway around a drive pulley with a radius of 0.4 meters. The contact length between the belt and the pulley is a semicircular arc: π(0.4), about 1.26 meters. Engineers use this contact length to calculate friction and power transmission. A longer contact arc generally means better grip between the belt and pulley.

Several common errors trip up students and professionals alike. Knowing them in advance helps you avoid them.

Mixing degrees and radians. This is the most frequent mistake. If your angle is 60 degrees but you plug it into s = rθ, you get 60r instead of the correct πr/3. Always check the angle unit before choosing your formula. When in doubt, convert to radians first.

Using diameter instead of radius. The formulas require the radius. If you are given a diameter of 14 cm, the radius is 7 cm. Forgetting to halve the diameter doubles your answer.

Forgetting measurement units. An arc length of 7.33 means nothing without units. If the radius was in centimeters, the arc length is in centimeters. Carry the units through every step.

Applying the wrong formula. The circle formulas s = rθ and s = πrθ/180 only work for circular arcs. For other curves, you need the calculus formula with the ∫. Using the circle formula on a parabola gives a wrong answer.

Confusing arc length with chord length. The chord is the straight line connecting the arc's endpoints. It is always shorter than the arc (except in trivial cases). If a problem asks for the curved distance, do not calculate the straight-line distance.

Rounding too early. Rounding intermediate values introduces error that compounds through the calculation. Keep full precision until the final step, then round to the requested number of decimal places.

Using incorrect integration limits. In the calculus formula, the limits must match the interval you are measuring. Reversing them flips the sign. Using limits from a different part of the problem gives the wrong segment.

Three related measurements describe different aspects of a circular region. Understanding each one prevents confusion.

Arc length measures the curved distance along the circle between two points. The formula is s = rθ with θ in radians. It answers the question: how far would you walk if you followed the curve?

Chord length measures the straight-line distance between the same two points. The formula is c = 2r sin(θ/2) with θ in radians. It answers the question: how far apart are the endpoints as the crow flies? The chord is always shorter than the corresponding minor arc.

Sector area measures the two-dimensional region bounded by two radii and the arc between them, like a pizza slice. The formula is A = (1/2)r²θ with θ in radians. It answers the question: how much area does the slice cover?

The comparison table below summarizes the differences.

Note that all three formulas require θ in radians. The sector area formula is especially easy to get wrong with degrees, because there is no simple degree version. Convert to radians first.

Try these problems, then check your answers below.

Problem 1 (degrees). A circle has radius 12 cm and a central angle of 45 degrees. Find the arc length.

Solution: Use s = πrθ/180 = π(12)(45)/180 = 3π ≈ 9.42 cm.

Problem 2 (radians). A circle has radius 8 m and a central angle of 2.5 radians. Find the arc length.

Solution: Use s = rθ = 8(2.5) = 20 m.

Problem 3 (diameter). A circle has diameter 20 inches and a central angle of 120 degrees. Find the arc length.

Solution: First convert diameter to radius: r = 20/2 = 10 inches. Then s = π(10)(120)/180 = 20π/3 ≈ 20.94 inches.

Problem 4 (missing radius). An arc of length 15 cm subtends a central angle of 0.75 radians. Find the radius.

Solution: Rearrange s = rθ to get r = s/θ = 15/0.75 = 20 cm.

Problem 5 (calculus). Find the arc length of y = (2/3)x^3/2 from x = 0 to x = 3.

Solution: The derivative is f'(x) = x^1/2, so [f'(x)]² = x. The ∫ is L = ∫[0 to 3] √(1 + x) dx. Let u = 1 + x, du = dx. When x = 0, u = 1. When x = 3, u = 4. L = ∫[1 to 4] √u du = (2/3)[u^3/2] from 1 to 4 = (2/3)(8 - 1) = 14/3 ≈ 4.67.

Arc length measures curved distance, whether along a circle or a general curve. For circles, use s = rθ with the angle in radians, or s = πrθ/180 with the angle in degrees. For other curves described by y = f(x), use the ∫ formula L = ∫√(1 + [f'(x)]²)dx over your interval. Watch out for degree-radian mix-ups, diameter-radius confusion, and premature rounding. The calculator at the top of this page handles all three modes with full step-by-step working. Enter your values, study the steps, and use the result with confidence.