Convert between radians and degrees

Learning Objectives Define a radian and explain its relationship to the radius and circumference of a circle. State the fundamental conversion relationship: π radians = 180°. Derive and apply the formula to convert angle measures from degrees to radians. Derive and apply the formula to convert angle measures from radians to degrees. Convert common angles (e.g., 30°, 45°, 60°, 90°, 180°) between degrees and radians without a calculator. Solve problems involving angle measures given in either radians or degrees. Ever wondered why your calculator has a 'RAD' mode next to the familiar 'DEG'? 🤔 It's not just an extra button; it's a gateway to a more natural way of measuring angles used in advanced math and science! While you're comfortable wit...

TermDefinitionExample Degree (°)A unit of angle measurement where one full rotation is divided into 360 equal parts. It is the most common way to measure angles in geometry.A right angle is 90°, a straight line is 180°, and a full circle is 360°. Radian (rad)The measure of a central angle of a circle that subtends an arc equal in length to the radius of that circle. It is a 'purer' measure because it is directly based on the properties of the circle itself.If a circle has a radius of 5 cm, a 1 radian angle cuts off an arc that is also 5 cm long. Unit CircleA circle with a radius of 1 unit, centered at the origin of a Cartesian plane. It is a fundamental tool in trigonometry because its circumference is 2π, directly linking radians to a full rotation.On a unit circle, a full rota...

The Fundamental Relationship \pi \text{ radians} = 180^{\circ} This is the cornerstone of all conversions. It states that a half-rotation of a circle is equivalent to π radians and 180 degrees. All conversion formulas are derived from this identity. Degrees to Radians Conversion \text{Radians} = \text{Degrees} \times \frac{\pi}{180^{\circ}} To convert an angle from degrees to radians, multiply the degree measure by the conversion factor π/180°. The degree units will cancel out, leaving you with radians. Radians to Degrees Conversion \text{Degrees} = \text{Radians} \times \frac{180^{\circ}}{\pi} To convert an angle from radians to degrees, multiply the radian measure by the conversion factor 180°/π. The π term will usually cancel out, leaving you with degrees.