Find The Measure Of The Arc Calculator

Arc Length Calculator

Calculate the measure of the arc (arc length) of a circle given its radius and central angle.

Common Angle Conversions

Degrees	Radians (Approx.)	Radians (Exact)
30°	0.5236	π/6
45°	0.7854	π/4
60°	1.0472	π/3
90°	1.5708	π/2
120°	2.0944	2π/3
180°	3.1416	π
270°	4.7124	3π/2
360°	6.2832	2π

Table showing the conversion of common angles from degrees to radians.

What is the Measure of the Arc (Arc Length)?

The measure of the arc, more commonly referred to as arc length, is the distance along the curved line making up the arc of a circle. Imagine walking along the edge of a circular pizza slice; the distance you walk along the crust is the arc length. It's a portion of the circle's circumference, determined by the central angle that subtends the arc and the radius of the circle. Our Arc Length Calculator helps you find this measure easily.

Anyone working with circular shapes or paths, such as engineers, architects, designers, mathematicians, and students, might need to calculate the measure of the arc. For example, it's used in designing curved roads, calculating the distance traveled by a point on a rotating wheel, or in various geometric problems.

A common misconception is that the "measure of the arc" refers to the central angle in degrees or radians. While the central angle *defines* the arc, the arc length is a distance, measured in units like centimeters, meters, or inches, just like the radius.

Arc Length Calculator Formula and Mathematical Explanation

The formula to find the measure of the arc (arc length) depends on whether the central angle is given in degrees or radians.

If the central angle θ is in radians, the formula is:

Arc Length (s) = r × θ

where:

• s is the arc length
• r is the radius of the circle
• θ is the central angle in radians

If the central angle θ is in degrees, we first convert it to radians by multiplying by π/180:

θ (radians) = θ (degrees) × (π / 180)

So, the formula becomes:

Arc Length (s) = r × (θ (degrees) × π / 180)

or

Arc Length (s) = (θ (degrees) / 360) × 2πr

This second form shows that the arc length is a fraction (θ/360) of the total circumference (2πr) of the circle. Our Arc Length Calculator uses the degrees input and converts it internally.

Variables Table

Variable	Meaning	Unit	Typical Range
r	Radius of the circle	Length (e.g., cm, m, inches)	Positive numbers (>0)
θdegrees	Central angle in degrees	Degrees (°)	0 to 360 (or more)
θradians	Central angle in radians	Radians (rad)	0 to 2π (or more)
s	Arc Length (Measure of the arc)	Length (same as radius)	Positive numbers (≥0)
C	Circumference	Length (same as radius)	Positive numbers (>0)

Practical Examples (Real-World Use Cases)

Example 1: Designing a Curved Path

An architect is designing a curved garden path that is part of a circle with a radius of 15 meters. The path covers a central angle of 60 degrees. What is the length of the path?

• Radius (r) = 15 m
• Central Angle (θ) = 60°

Using the Arc Length Calculator or the formula:

Angle in radians = 60 × (π / 180) = π/3 radians ≈ 1.047 radians

Arc Length = 15 m × (π/3) ≈ 15 × 1.047 ≈ 15.708 meters.

The garden path will be approximately 15.71 meters long.

Example 2: Ferris Wheel Travel

A Ferris wheel has a radius of 50 feet. If a cart travels through an angle of 135 degrees, how far did it travel along the circumference?

• Radius (r) = 50 ft
• Central Angle (θ) = 135°

Using the Arc Length Calculator:

Angle in radians = 135 × (π / 180) = 3π/4 radians ≈ 2.356 radians

Arc Length = 50 ft × (3π/4) ≈ 50 × 2.356 ≈ 117.8 feet.

The cart traveled approximately 117.8 feet.

How to Use This Arc Length Calculator

1. Enter the Radius (r): Input the radius of the circle in the first field. Ensure it's a positive number.
2. Enter the Central Angle (θ): Input the central angle subtended by the arc, in degrees, in the second field.
3. Calculate: The calculator automatically updates the results as you type or after you click the "Calculate" button.
4. Read the Results:
   • Arc Length: The primary result shows the calculated measure of the arc in the same units as the radius.
   • Angle in Radians: The central angle converted to radians.
   • Circumference: The total circumference of the circle.
5. Visual Representation: The chart below the calculator visually represents the sector and arc based on your inputs.
6. Reset: Use the "Reset" button to return to default values.
7. Copy Results: Use the "Copy Results" button to copy the input values and results to your clipboard.

This Arc Length Calculator is a straightforward tool for quickly finding the measure of the arc.

Key Factors That Affect the Measure of the Arc

1. Radius (r): The larger the radius of the circle, the longer the arc length will be for the same central angle. It's a direct proportional relationship.
2. Central Angle (θ): The larger the central angle, the longer the arc length will be for the same radius. The arc length is directly proportional to the central angle.
3. Units of Radius: The unit of the arc length will be the same as the unit used for the radius (e.g., if the radius is in meters, the arc length will be in meters).
4. Units of Angle: Our calculator specifically asks for the angle in degrees. If you have the angle in radians, you'd use the simpler formula s = r × θ_radians, or you could convert radians to degrees first (degrees = radians × 180/π).
5. Accuracy of π: The value of Pi (π) used in the calculation affects precision. Most calculators and our tool use a high-precision value of π.
6. Measurement Precision: The accuracy of your input values for radius and angle will directly impact the accuracy of the calculated arc length.

Understanding these factors helps in both using the Arc Length Calculator and interpreting its results for finding the measure of the arc.

Frequently Asked Questions (FAQ)

Q1: What is the difference between arc length and the measure of an arc?

A1: Often, "measure of an arc" can refer to the central angle in degrees, but when talking about the *length* of the arc, it's the distance along the curve. This calculator finds the arc length, which is a distance.

Q2: Can the central angle be greater than 360 degrees?

A2: Yes. If the angle is greater than 360 degrees, it means the arc wraps around the circle more than once. Our Arc Length Calculator can handle these values.

Q3: What units are used for arc length?

A3: The arc length will be in the same units as the radius you input (e.g., meters, feet, inches).

Q4: How do I convert an angle from radians to degrees?

A4: To convert radians to degrees, multiply the angle in radians by 180/π.

Q5: How do I find the arc length if I only know the chord length and radius?

A5: If you have the chord length (c) and radius (r), you can first find the central angle θ using the formula c = 2r * sin(θ/2). Then solve for θ and use the arc length formula. You might need our Angle Measure Tool or other geometry calculators for this.

Q6: Can I use this calculator for any circle?

A6: Yes, this Arc Length Calculator works for any perfect circle or sector of a circle, provided you know the radius and the central angle.

Q7: What if my angle is in radians?

A7: This calculator specifically takes degrees. You can convert your radian angle to degrees (degrees = radians * 180/π) before inputting, or use the direct formula s = r * θ_radians manually.

Q8: Is arc length related to sector area?

A8: Yes, both arc length and sector area depend on the radius and central angle. The sector area is (θ/360) * πr² (for θ in degrees).