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Angle Unit Converter

radians equals degrees times pi over 180

Angle Conversion =

1 Degree (°) = 0.017453293 Radian (rad)

Solution in Other Units

UnitValue
Degree (°)1
Radian (rad)0.017453293

Show Your Work

Result = Value × (Source factor ÷ Target factor)
Base unit for this angle family is Degree (°).
1 Degree (°) = 1 Degree (°)
1 Degree (°) = 0.017453293 Radian (rad)
See the full table below for all angle conversions.
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How It Works

This converter handles degree and radian measure. Degrees divide a full turn into 360 equal parts, while radians measure angle from arc length divided by radius. The calculator converts your source angle using the exact degree-radian relationship and then displays the equivalent value in the other unit.

Example Problem

Convert 120 degrees to radians for a trigonometry problem.

  1. Start with the source value: 120°.
  2. Use the formula radians = degrees × π / 180.
  3. Substitute the value: 120 × π / 180.
  4. Simplify 120 / 180 to 2 / 3.
  5. So the result is 2π / 3 radians.
  6. As a decimal, that is about 2.0944 radians.

Key Concepts

Angles can be expressed in degrees or radians. Degrees are familiar from geometry and navigation, while radians are the natural unit for calculus, physics, and most programming-language trigonometric functions. Understanding both forms is essential because many formulas are simplest in radians, but many people reason more comfortably in degrees.

Applications

  • Mathematics and trigonometry: switching between classroom degree notation and radian-based identities
  • Programming and graphics: converting user-facing degree inputs into the radian values expected by many libraries
  • Physics and engineering: interpreting angular position, phase, and oscillation formulas written in radians
  • CAD and geometry work: checking whether an angle shown on a diagram matches a formula-driven result

Common Mistakes

  • Sending degree values into trig functions that expect radians, which can produce wildly incorrect results
  • Rounding π too early when you need a cleaner symbolic result such as π/4 or 2π/3
  • Forgetting that a full circle is 360° but only 2π radians

Frequently Asked Questions

How do you convert angles between degrees and radians?

Multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.

What is the formula for converting degrees to radians?

Radians = Degrees × π / 180.

How many radians are in a full circle?

A full circle is 2π radians.

How many degrees are in π radians?

π radians equals 180 degrees.

Why do math and physics formulas prefer radians?

Radians make many derivatives, identities, and periodic formulas cleaner and more natural.

Should I use degrees or radians in calculator apps and code?

Use whichever unit the formula or software expects, and convert if necessary.

Can this converter help with trig homework and programming?

Yes. It is useful whenever you need to move cleanly between human-readable degree values and radian-based formulas.

Angle Conversion Formula

Angle conversions relate degree measure to radian measure. The calculator converts your source angle into the common internal representation first, then reports the equivalent angle in the other supported unit below.

Result = Value × (Source factor ÷ Target factor)
  • Result — the converted measurement in the target angle unit
  • Value — the original measurement you enter
  • Source factor — the factor that maps the source unit to the common base unit
  • Target factor — the factor used to express the same base-unit value in the target unit

Worked Examples

Geometry

How do you convert 45 degrees to radians?

A diagram is labeled in degrees, but the formula you need expects radians.

  • Start with the source value: 45°.
  • Use the formula radians = degrees × π / 180.
  • Compute 45 × π / 180 = π / 4.
  • As a decimal, π / 4 ≈ 0.785398 radians.
  • That is the radian equivalent of 45°.
  • So 45 degrees is about 0.7854 radians.

An angle of 45° is about 0.7854 radians.

This is the classic quarter-of-a-right-angle conversion used all over trigonometry.

Trig Functions

What is 0.5 radians in degrees?

A software library reports an angle in radians, but you want a more intuitive degree value.

  • Start with the source value: 0.5 rad.
  • Use the formula degrees = radians × 180 / π.
  • Compute 0.5 × 180 / π ≈ 28.6479°.
  • This is a little under 30 degrees.
  • The same angle is about 28.65°.
  • So 0.5 radians equals about 28.65 degrees.

An angle of 0.5 rad is about 28.65°.

This is a handy conversion when checking radians-based outputs from code or calculators.

Rotation Checks

How do you convert π radians to degrees?

A formula produces π radians and you want the corresponding degree measure for a diagram.

  • Start with the source value: π radians.
  • Recall that π radians equals half a full turn.
  • Half of 360° is 180°.
  • You can also verify it with degrees = radians × 180 / π.
  • π × 180 / π = 180°.
  • So π radians equals 180 degrees.

An angle of π radians equals 180°.

Recognizing common radian angles makes it easier to spot-check trig answers quickly.

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