Degree to Radian Converter (convert ° to rad)

Convert an angle from degrees to radians for mathematics, engineering, programming, navigation, or calculations involving circles and rotation.

180° = π rad

Degrees and Radians Explained

Degrees and radians are two ways of measuring the same angle. Degrees divide one complete turn into 360 equal parts, while radians measure an angle using the radius and arc of a circle.

• 90° = π/2 rad, or a quarter turn
• 180° = π rad, or a half turn
• 360° = 2π rad, or one complete turn

Degrees are familiar from weather vanes, maps, compasses, and everyday descriptions of rotation. Radians are commonly used in trigonometry, physics, engineering, computer graphics, and calculus.

Degree to Radian Conversion Formula

One complete turn is equal to 360 degrees or 2π radians:

360° = 2π rad

Dividing both sides by two gives the relationship used for conversion:

180° = π rad

To convert degrees to radians, multiply the angle by π and divide by 180:

Radians = Degrees × π ÷ 180

You can also use a decimal conversion factor:

Radians ≈ Degrees × 0.01745329252

How to Convert Degrees to Radians

Follow these three steps:

1. Write down the angle in degrees.
2. Multiply it by π/180.
3. Simplify the resulting fraction where possible.

Example: Convert 60° to Radians

Multiply 60 by π/180:

60° × π/180 = 60π/180 = π/3 rad

As a decimal, π/3 is approximately 1.0472 radians.

Example: Convert 225° to Radians

225° × π/180 = 225π/180 = 5π/4 rad

As a decimal, 5π/4 is approximately 3.9270 radians.

Exact and Decimal Answers

An exact radian value is normally written using π. For example, the exact form of 45° is π/4 radians.

A calculator may instead show the decimal value, approximately 0.785398 radians. Both describe the same angle, but the expression containing π is exact.

When completing a longer calculation, keep the exact value for as long as possible and round only the final answer.

Degree to Radian Conversion Table

From Degrees to Radians: How Angles Guide a Robotic Arm

Consider a robotic arm transferring boxes from a conveyor to a pallet. Its shoulder, elbow and wrist must rotate to carefully coordinated positions before the gripper can collect each box. Those joint positions may be displayed to a technician in degrees, while the robot’s programming interface or mathematical model represents them in radians.

For one part of the movement, the target positions might be:

• Shoulder: 45°
• Elbow: 90°
• Wrist: −30°

The signs indicate direction according to the coordinate convention configured for each joint. Positive does not always mean clockwise: the result depends on the joint axis and the direction from which it is viewed.

1. A Target Position Is Taught or Programmed

During setup, a technician may move the robot into position with a teaching pendant or enter coordinates through programming software. Degrees are convenient for human-readable displays because quarter, half and full turns correspond to the familiar values of 90°, 180° and 360°.

The operator may specify individual joint positions, but many industrial tasks are programmed by teaching the required location and orientation of the tool. In that case, the controller uses robot kinematics to determine which combination of joint angles places the gripper at the requested pose.

Before movement begins, the control system checks whether the target is reachable and whether the proposed joint positions remain within the configured limits. Real safety behaviour depends on the robot model, controller settings and installed safeguards.

2. The Software Uses the Required Angle Unit

Degrees and radians describe the same rotation, but their numerical values differ. When a software interface requires radians, a value entered in degrees must be converted with:

Radians = degrees × π ÷ 180

For the elbow:

90° × π ÷ 180 = π/2 rad ≈ 1.5708 rad

The required unit must be confirmed from the relevant software documentation rather than assumed. For example, NVIDIA’s Isaac Sim articulation-controller documentation specifies radians for angular commands while noting that angles in its USD scene representation use degrees. Other robots and development tools may permit the unit to be selected.

3. The Controller Builds a Motion Rather Than Jumping to the Target

A target angle is not normally sent to a motor as an instruction to move instantly. The motion controller generates a trajectory between the robot’s present configuration and its destination. That trajectory defines how joint position, velocity and acceleration should change throughout the move.

The joints must be coordinated because moving one joint changes the position of every link farther along the arm. If the shoulder, elbow and wrist followed unrelated timing profiles, the gripper could take an unintended route even if every joint eventually reached the correct final angle.

A joint-space movement coordinates the joint variables directly. A Cartesian movement instead attempts to guide the tool along a specified path, requiring the controller to calculate changing joint positions along that path.

4. Joint Angles Are Translated into Actuator Commands

The controller transforms the planned motion into commands suitable for the robot’s actuators. Depending on the system, this may involve target positions, angular velocities, torque commands, encoder counts or other hardware-level values.

Joint sensors report the arm’s measured state. A feedback controller compares those measurements with the planned trajectory and adjusts the actuator commands to reduce the error. This process runs repeatedly as the arm accelerates, travels and comes to rest.

Robotics interfaces vary in the quantities they expose. Boston Dynamics, for example, documents joint positions in radians, joint velocities in radians per second and loads in newton-metres in its joint-control API . ABB also provides a RobotStudio example in which angular values supplied in radians are converted to degrees for readable output.

What Happens If Degrees Are Mistaken for Radians?

A value of 90 degrees represents one quarter of a revolution, whereas 90 radians represents approximately 14.32 revolutions:

90 rad ÷ 2π ≈ 14.32 revolutions

This does not mean a well-designed industrial robot will necessarily spin a joint more than fourteen times. Joint limits, range checks, angle normalization and safety systems may reject or transform the command. However, a unit mismatch can still produce an invalid target, an incorrect simulated path or a movement very different from the one intended.

Unit information should therefore travel with every angular value. Variable names such as elbowDegrees and elbowRadians, explicit conversion functions and automated tests can make mistakes easier to detect.

Why Radians Are Useful in Robotics

Radians connect an angle directly to the geometry of a circle. When an angle is expressed in radians, the arc length is:

Arc length = radius × angle

This relationship makes radians especially convenient in trigonometry, calculus, angular velocity, kinematics and dynamics. Degrees may remain easier for a person to interpret, but radians often produce cleaner mathematical relationships inside engineering software.

Safe Testing Comes Before Production Movement

A degree-to-radian conversion is only one small part of commanding a robot safely. A real installation must also account for payload, collisions, singularities, speed limits, tool dimensions, work-cell boundaries and the manufacturer’s operating procedures.

New motions should be validated using the approved simulation, commissioning and reduced-speed procedures for the particular robot. Physical guarding, safety-rated controls and emergency-stop equipment must not be replaced by software checks alone.