Rad to Angle: Degrees & Radians Conversion

This calculator converts an angle given in radians to degrees, and also shows the equivalent in degrees, minutes, and seconds. You enter the radian value (a number, often involving π), and the calculator returns the angle in degrees and in degrees-minutes-seconds format. The formula is simple: multiply radians by 180/π.

Formula

The conversion from radians to degrees uses the fundamental relationship that a full circle is both 2π radians and 360°. Therefore:

Degrees = Radians × (180 / π)

Where:

• Radians – the angle you want to convert (unit: rad)
• Degrees – the resulting angle (unit: °)
• π – approximately 3.141592653589793

For the degrees-minutes-seconds (DMS) output, the decimal part of the degrees is multiplied by 60 to get minutes, and the decimal part of the minutes is multiplied by 60 to get seconds.

Worked Example

Using the default value of 1.5708 radians (which is approximately π/2):

1. Multiply by 180/π: 1.5708 × (180 / 3.141592653589793) = 90.0000°
2. The integer part is 90°, the decimal is 0.0000, so minutes and seconds are 0′ 0″.

The calculator shows 90.0000° and 90° 0′ 0″.

Why Convert Radians to Degrees?

Radians are the natural unit of angle in mathematics and physics because they simplify many formulas. For example, the derivative of sin(x) is cos(x) only when x is in radians. However, most people think in degrees for everyday angles – a right angle is 90°, a full turn is 360°. When you read a problem in a textbook, use a calculator with a degree mode, or set a machine tool, you often need to switch from radians to degrees.

Common situations where you need this conversion:

• Solving trigonometry problems where the answer is in radians but you need degrees for a drawing or a physical angle.
• Converting angular velocity (rad/s) to revolutions per minute (RPM) – first to degrees, then divide by 360.
• Using GIS or mapping software that stores angles in radians but displays them in degrees.
• Programming graphics or games where rotation is often specified in radians but you want to report degrees.

How to Convert Manually

If you don’t have a calculator, you can still do the conversion by hand:

1. Write down the radian value. For example, 2.5 rad.
2. Multiply it by 180. Example: 2.5 × 180 = 450.
3. Divide by π (≈3.14159). Example: 450 / 3.14159 = 143.239°.

If your radian value includes π, like 3π/4, you can cancel π: 3π/4 × 180/π = (3 × 180) / 4 = 135°. This shortcut works when the radian value is a simple multiple of π.

Common Mistakes and Pitfalls

The most frequent error is forgetting to multiply by 180/π and instead multiplying by π/180 (which converts degrees to radians). Always check: if you start with a number less than 6.2832 (2π), the degree result should be less than 360. If you get a huge number, you likely used the wrong factor.

Another mistake is rounding π too early. Using 3.14 instead of 3.14159 can change the result by a few hundredths of a degree. For precise work, keep π to at least 5 decimal places or use the calculator’s built-in value.

Also, be careful with negative angles. A negative radian value converts to a negative degree value – for example, -1 rad ≈ -57.2958°. This is correct and often used in mathematics to indicate direction (clockwise vs counterclockwise).

Degrees, Minutes, and Seconds (DMS)

In navigation, surveying, and astronomy, angles are often expressed in degrees, minutes, and seconds. One degree is divided into 60 minutes (symbol ′), and one minute into 60 seconds (symbol ″). For instance, 30.5° is 30° 30′ 0″.

To convert a decimal degree value to DMS:

1. The integer part is the degrees.
2. Multiply the decimal part by 60 – the integer part is the minutes.
3. Multiply the remaining decimal by 60 – that gives the seconds.

For example, 143.239° becomes 143° (0.239 × 60 = 14.34′), then 0.34 × 60 = 20.4″, so 143° 14′ 20″.

This format is useful when you need to enter coordinates into a GPS or read a survey map.

Limitations of the Conversion

This conversion is purely mathematical and always exact, but the precision of the output depends on the input. If you enter a radian value with only a few decimal places, the degree result will be approximate. For example, entering 1.5708 (a rounded version of π/2) gives 90.0000°, but the exact value of π/2 gives exactly 90°. The calculator shows four decimal places by default, which is sufficient for most practical purposes.

Also, note that radians are dimensionless – they are a ratio of arc length to radius – so the unit “rad” is often omitted. When you see an angle without a unit in higher mathematics, it is usually in radians.

Frequently Asked Questions

How many degrees in one radian?

One radian is exactly 180/π degrees, which is approximately 57.295779513°. That means a little more than 57 degrees in one radian.

What is 2π radians in degrees?

2π radians is exactly 360°, a full circle. This is the most important conversion to remember: 2π rad = 360°, so π rad = 180°, and π/2 rad = 90°.

How do I convert degrees to radians?

Multiply the degree value by π/180. For example, 45° × π/180 = π/4 radians. This is the inverse of the radian-to-degree conversion.

Why use radians instead of degrees?

Radians simplify calculus and many physics equations because they are a natural unit based on the circle’s radius. For instance, the derivative of sin(x) is cos(x) only when x is in radians. In engineering and everyday measurements, degrees are often more intuitive.

Can I enter π as a number?

Yes, you can type the decimal approximation of π, like 3.14159, or use the calculator’s built-in π constant if available. For exact results, it’s better to use the symbol π in your own calculations, but for a numeric calculator, use 3.141592653589793.

What is the difference between a radian and a steradian?

A radian measures a planar angle (2D), while a steradian measures a solid angle (3D). One steradian is the solid angle that subtends an area of r² on a sphere’s surface. They are related but not directly convertible.

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in r | Radians | rad | 1.5708 | 0
out deg | Degrees | ° | r * 180 / pi | 4
out dms | Degrees, minutes, seconds | ° ' " | (r * 180 / pi) | 0