Angle Converter

Introduction

Angles describe rotation: how far one ray, line, or direction turns relative to another. They show up everywhere—geometry homework, compass bearings, camera fields of view, robotics joints, and the tiny separations between stars. Different fields adopted different angle units over time, so it’s common to receive an angle in one unit (for example, degrees) while your calculator, software, or specification expects another (often radians).

This page converts a single input angle into five common units: degrees (°), radians (rad), gradians (grad), arcminutes (′), and arcseconds (″). The conversion runs entirely in your browser, so your input stays on your device.

How to use the angle converter

1. Enter a numeric value in the Value field (decimals are allowed).
2. Select the unit that value is currently in (Degrees, Radians, Gradians, Arcminutes, or Arcseconds).
3. Select Convert to see the equivalent values in all supported units.

Tip: If you are working with latitude/longitude or navigation bearings, remember that 1 degree = 60 arcminutes and 1 arcminute = 60 arcseconds. This converter outputs decimal values for each unit, which is often the most convenient format for spreadsheets, CAD tools, and programming.

Formula and conversion relationships

The converter uses a simple approach: it first converts the input to a base value in degrees, then derives all other units from that base. This reduces the chance of mistakes compared with chaining multiple conversions. The key relationships are:

• Degrees ↔ Radians: rad = deg × (π / 180), and deg = rad × (180 / π)
• Degrees ↔ Gradians: 400 grad = 360°, so 1 grad = 0.9° and grad = deg / 0.9
• Degrees ↔ Arcminutes: 1° = 60′, so arcmin = deg × 60 and deg = arcmin / 60
• Degrees ↔ Arcseconds: 1° = 3600″, so arcsec = deg × 3600 and deg = arcsec / 3600

In mathematical notation, the degrees-to-radians relationship is:

Worked example

Convert 30 degrees to the other units.

• Radians: 30 × (π / 180) = π/6 ≈ 0.5235987756 rad
• Gradians: 30 / 0.9 = 33.3333333333 grad
• Arcminutes: 30 × 60 = 1800′
• Arcseconds: 30 × 3600 = 108000″

If you enter 30 and choose Degrees (°), the results table below will show these same values.

The many ways to measure angles (background)

The degree is the most familiar unit for everyday use. A full circle contains 360 degrees, a convention often linked to ancient Babylonian astronomy and base‑60 arithmetic. Degrees are subdivided into arcminutes and arcseconds: 1° = 60′ and 1′ = 60″. This sexagesimal structure mirrors timekeeping (hours, minutes, seconds) and remains common in navigation and mapping.

The radian is the natural unit in mathematics. It is defined by the geometry of a circle: one radian is the angle that subtends an arc equal in length to the radius. A full rotation is 2π radians, so 1 rad ≈ 57.2958°. Radians are standard in calculus and most programming libraries because many trigonometric identities and derivatives take their simplest form in radians.

The gradian (also called gon) divides a circle into 400 parts, so a right angle is exactly 100 grad. It was promoted during the French Revolution as a metric-friendly alternative. While not universal, it still appears in some surveying and engineering contexts, especially when reading older documents.

Arcminutes and arcseconds are used when angles are very small. Astronomers describe apparent sizes and separations in arcminutes and arcseconds; optical engineering and vision science also use these units. Because the numbers can be large (for example, 1° = 3600″), quick conversion helps prevent mistakes.

Quick reference conversion table

The table below summarizes common conversion factors. It can be used to sanity-check results (for example, 180° should always equal π radians).

Limitations and assumptions

• Decimal output: Results are shown as decimal numbers. Some angles are commonly expressed as exact multiples of π (for example, π/2), but this tool does not attempt symbolic simplification.
• Floating-point precision: Calculations use JavaScript floating-point numbers. This is accurate for typical use, but extremely large values may show rounding artifacts.
• Input validation: The form requires a number, but it does not enforce a specific range. Negative angles and angles greater than one full rotation are allowed and will be converted normally.
• Arcminutes/arcseconds format: The converter treats arcminutes and arcseconds as standalone units (decimal minutes/seconds), not as a degrees-minutes-seconds (DMS) combined format.

If you need DMS formatting (e.g., 12° 34′ 56″) or normalization to a specific range (such as 0–360°), you can still use this converter as a first step by converting to degrees and then formatting the result.

Practical guidance: choosing the right unit

When you are deciding which unit to use, it helps to match the unit to the task. In everyday contexts, degrees are easiest to read because most people have an intuitive sense of what 10°, 45°, or 180° looks like. In contrast, radians are often the best choice for calculations because they connect angle directly to arc length: if a circle has radius r and an angle of θ radians, the arc length is s = rθ. That simple relationship is one reason physics and engineering formulas frequently assume radians.

Gradians are common in some surveying workflows because a right angle is exactly 100 grad, which can make certain field calculations and instrument readouts feel more “decimal.” Arcminutes and arcseconds are best when you need to describe small angles precisely, such as the apparent diameter of the Moon (about 30 arcminutes) or the resolution of an optical system. If you are comparing very small differences, expressing everything in arcseconds can avoid lots of leading zeros.

Common conversions and sanity checks

A few anchor values are worth memorizing because they help you catch unit mix-ups quickly. If your result is far from these reference points, you may have selected the wrong input unit.

• 90° = π/2 rad = 100 grad = 5400′ = 324000″
• 180° = π rad = 200 grad = 10800′ = 648000″
• 360° = 2π rad = 400 grad = 21600′ = 1296000″
• 1 rad ≈ 57.2958° (a little more than 57°)
• 1° ≈ 0.0174533 rad (about 0.0175 rad)

Another quick check: because 400 grad equals 360°, a gradian is slightly smaller than a degree. That means a number in gradians should be slightly larger than the same angle in degrees (for example, 45° equals 50 grad). If you see the opposite, the conversion direction may be reversed.

Notes for navigation, mapping, and astronomy

In navigation and mapping, angles often appear as bearings, headings, or coordinates. Bearings are typically expressed in degrees, sometimes with minutes and seconds. Many GIS tools and APIs store coordinates in decimal degrees, but some datasets still use DMS. If you are converting a DMS value manually, remember that minutes and seconds are fractions of a degree: deg + (min/60) + (sec/3600). This calculator can help once you have a single decimal value.

In astronomy, arcminutes and arcseconds are especially common because celestial objects can be separated by tiny angles. For example, the full Moon is roughly 0.5° across, which is about 30′ or 1800″. A typical “good seeing” night at an observatory might be described as 1″ to 2″ of atmospheric blur. Converting between degrees and arcseconds is therefore a routine step when comparing telescope specifications, camera pixel scales, and star catalog data.

FAQ

Does the converter accept negative angles?

Yes. Negative values represent rotation in the opposite direction (for example, clockwise instead of counterclockwise, depending on your convention). The converter applies the same unit relationships to negative numbers.

Can I convert angles larger than one full turn?

Yes. Angles greater than 360° (or greater than 2π radians) are valid in many contexts, such as accumulated rotation. This tool does not automatically wrap or normalize the result; it simply converts the magnitude you enter.

Why do my results show many decimal places?

Some conversions involve π, which is irrational, so the decimal representation never terminates. JavaScript uses floating-point arithmetic, so the output is a decimal approximation. If you need a rounded value, you can round the displayed results for your use case (for example, to 3 or 6 decimal places).

Is an arcminute the same as a minute of time?

No. An arcminute is an angular unit (1/60 of a degree). A minute of time is a unit of duration. The names are similar because both use base-60 subdivisions, but they measure different things.

What is the difference between gradians and degrees?

Both measure the same geometric quantity, but they divide the circle differently. Degrees divide a circle into 360 parts; gradians divide it into 400. As a result, 1 grad = 0.9°, and 1° = 10/9 grad.

Summary

Use the form below to convert between degrees, radians, gradians, arcminutes, and arcseconds. The calculator is designed for quick, reliable unit conversion: enter a value, select the unit, and the results table will show all equivalent values. For best results, double-check your input unit and use the reference values above to confirm the output is in the expected range.