Arcsecond — Definition, Formula & Examples

An arcsecond is a tiny unit of angle measurement equal to one-sixtieth of an arcminute, or equivalently one three-thousand-six-hundredth of a degree. It is used when extremely small angles need to be described precisely.

An arcsecond (symbol: ″ or ") is defined as $\frac{1}{3600}$ of one degree of arc, where one full rotation equals $360°$ . Equivalently, $1'' = \frac{1}{60}$ arcminute $= \frac{\pi}{648{,}000}$ radians.

Key Formula

\theta_{\text{arcsec}} = \theta_{\text{deg}} \times 3600

Where:

$\theta_{\text{arcsec}}$ = Angle measured in arcseconds
$\theta_{\text{deg}}$ = Angle measured in decimal degrees
$3600$ = Number of arcseconds in one degree (60 arcminutes × 60 arcseconds)

How It Works

Degrees can be subdivided just like hours on a clock. One degree splits into 60 arcminutes (′), and each arcminute splits into 60 arcseconds (″). So there are

60 \times 60 = 3600

arcseconds in a single degree. To convert from decimal degrees to arcseconds, multiply by 3600. To go the other way, divide the number of arcseconds by 3600.

Worked Example

Problem: Convert an angle of 0.015° to arcseconds.

Step 1: Multiply the angle in degrees by 3600 to convert to arcseconds.

0.015 \times 3600 = 54''

Step 2: You can verify by expressing this in arcminutes first: 0.015° × 60 = 0.9′, and 0.9′ × 60 = 54″.

Answer: 0.015° equals 54 arcseconds.

Why It Matters

Arcseconds are essential in astronomy, where the apparent sizes of stars and planets are fractions of a degree. Surveyors and GPS engineers also rely on arcsecond precision — on Earth's surface, one arcsecond of latitude corresponds to roughly 31 meters, so even a small angular error translates into real distance.

Common Mistakes

Mistake: Confusing arcseconds (angular measure) with seconds (time).

Correction: An arcsecond measures angle, not time. The word 'arc' distinguishes it. Use the double-prime symbol (″) for arcseconds and 's' for seconds of time.