What Is Arc Length?
Arc length is the distance measured along the
curved portion of a circle. Unlike a straight-line
distance, arc length follows the circumference
between two points on the circle.
The arc length depends on two main measurements:
the radius of the circle and the central angle
formed by the arc.
A larger radius produces a longer arc for the same
angle, while a larger central angle also produces
a longer arc for the same radius.
This Arc Length Calculator lets you calculate the
curved distance using either degrees or radians.
How to Use the Arc Length Calculator
Enter the radius and central angle of your circle,
select the angle unit, and the calculator will
instantly determine the arc length.
1
Enter the Radius
Enter the distance from the center of the
circle to its circumference.
2
Enter the Angle
Enter the central angle and choose whether
it is measured in degrees or radians.
3
Get the Result
Click the calculate button to see the arc
length and additional circle measurements.
Arc Length Formulas
Arc Length Using Radians
When the central angle is expressed in radians,
the simplest arc length formula is:
L = rθ
In this formula, L is the arc length, r is the
radius, and θ is the central angle in radians.
Arc Length Using Degrees
If the central angle is given in degrees, use the
fraction of the complete 360-degree circle:
L = (θ / 360) × 2πr
For example, a 90-degree arc represents one-quarter
of the circumference of the complete circle.
Arc Length Calculation Example
Suppose a circle has a radius of
10 units and a central angle
of 90°.
Using the degrees formula:
L = (θ / 360) × 2πr
L = (90 / 360) × 2π(10)
L = 5π
L ≈ 15.71 units
Therefore, the arc length is approximately
15.71 units.
Because 90° is one-quarter of a complete circle,
the arc is also one-quarter of the circle's
circumference.
Degrees vs. Radians for Arc Length
Arc length calculations can use either degrees or
radians, but the formula changes depending on the
angle unit.
Degrees:
L = (θ / 360) × 2πr
Radians:
L = rθ
To convert degrees to radians, multiply the angle
by π/180. To convert radians to degrees, multiply
the angle by 180/π.
Radians = Degrees × π / 180
Degrees = Radians × 180 / π
Where Is Arc Length Used?
Arc length is commonly used in geometry,
trigonometry, engineering, construction, design,
surveying, and many practical applications
involving circular objects.
It can be useful when calculating curved paths,
circular tracks, machine components, wheels,
pipes, architectural structures, and portions of
circular boundaries.
Students can also use arc length calculations to
practice relationships between radius, central
angles, circumference, and radians.
Frequently Asked Questions
What is the formula for arc length?
When the angle is in radians, the formula is
L = rθ. When the angle is in degrees, use
L = (θ / 360) × 2πr.
What is arc length?
Arc length is the distance measured along the
curved portion of a circle between two points.
Can I calculate arc length using degrees?
Yes. Use L = (θ / 360) × 2πr when the central
angle is measured in degrees.
Can I calculate arc length using radians?
Yes. When the angle is in radians, simply multiply
the radius by the central angle: L = rθ.
What happens to arc length when the radius increases?
For the same central angle, arc length increases
proportionally with the radius.
What is the arc length of a 90-degree arc?
A 90-degree arc is one-quarter of the complete
circumference, so its length is πr/2.
What is the arc length of a semicircle?
A semicircle has a 180-degree central angle.
Its curved arc length is πr.
What is the difference between arc length and circumference?
Circumference is the length around the entire
circle. Arc length measures only a selected
portion of that circular boundary.
How do I convert degrees to radians?
Multiply the angle in degrees by π/180.
For example, 180° equals π radians.
Can arc length be greater than the circumference?
For a standard single arc with a central angle
from 0° to 360°, the arc length cannot exceed
the full circumference.