Seconds To Radians Converter

second to radian – How to convert ″ to rad

When you’re working at ultra-small angular scales, especially in astronomy, satellite systems, or surveying, angles are often measured in arcseconds (″). But for calculations in physics, engineering, and mathematics, those values need to be in radians, the default unit used in formulas and simulations.

This conversion might seem like a niche detail, but it plays a big role in making real-world measurements compatible with mathematical models.

What is a second (″) of arc?

An arcsecond, or simply second, is a unit of angular measurement in the sexagesimal system. It’s a subdivision of a degree:

• 1 degree = 60 arcminutes (′)

• 1 arcminute = 60 arcseconds (″)

• So, 1 degree = 3,600 arcseconds

Arcseconds are used where extreme angular precision is required. You’ll find them in:

• Astronomical coordinates and telescope tracking

• Geodetic surveying and GPS mapping

• Military-grade optical targeting

• Satellite communication and alignment

In many of these fields, 1 arcsecond of error can translate into significant physical deviations over distance.

What is a radian (rad)?

A radian is the SI unit for angular measurement and the standard in trigonometry, calculus, and rotational mechanics.

• 1 radian is the angle formed when an arc length equals the radius of the circle

• 1 full circle = 2π radians

• 1 radian ≈ 57.2958 degrees

Radians are used in all kinds of formulas—from computing torque and angular acceleration to analyzing signal phase and wave motion.

So when your input data comes in arcseconds, converting to radians is essential before applying it to any math-based model.

How to convert second to radian

Here’s the logic behind the conversion:

• 1 degree = 3,600 arcseconds

• 1 degree = π / 180 radians

• So, 1 arcsecond = (π / 180) ÷ 3,600 radians

This simplifies to:

1 arcsecond = π / 648,000 radians

Formula

radian = second × (π / 648,000)
Or approximately:
radian ≈ second × 0.00000484814

This is your go-to formula for precise arcsecond-to-radian conversion.

Example: Convert 5,000 arcseconds to radians

radian = 5,000 × (π / 648,000)
radian ≈ 5,000 × 0.00000484814
radian ≈ 0.02424

So, 5,000 arcseconds equals approximately 0.02424 radians.

To speed things up, use the Angle Converter or explore other categories in our full set of Conversion tools.

Where a fraction of a degree means everything

In astrophysics, even the smallest angular miscalculation can result in significant error when mapping star systems or tracking celestial bodies. A shift of just a few arcseconds could mean missing the transit of a planet or losing telescope lock on a deep-space object.

In geolocation, arcsecond-level resolution can define the difference between precise, on-target GPS guidance and a several-meter miss—critical in aviation, defense, and emergency response.

And in robotics, optical tracking systems sometimes rely on sub-degree accuracy. The source may give data in arcseconds, but the control system requires radians. Without conversion, even slight discrepancies can cascade into larger movement errors.

Precision starts with the right units

Arcseconds are built for fine-tuned observation. Radians are built for formulas. If you're moving between measurement and computation, converting second to radian keeps your data aligned with the tools that rely on it.

Quick recap:

radian = second × (π / 648,000)
or
radian ≈ second × 0.00000484814

Use the Angle Converter to handle this conversion instantly, or check out more options in our full Conversion tools catalog.