Radian Measure of Angles
Aligned to F-TF.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
A radian is the angle that cuts off an arc equal in length to the radius. On the unit circle a full turn is 2π radians, so 180° equals π radians. Convert degrees to radians by multiplying by π/180.
Worked examples
Example 1
Convert 60° to radians. Multiply by π/180: 60 × π/180 = 60π/180 = π/3 radians. So a 60° angle subtends an arc of length π/3 on the unit circle.
Example 2
Convert 3π/4 radians to degrees. Multiply by 180/π: (3π/4)(180/π) = 3(180)/4 = 135°. A useful anchor: π radians is a half turn, 180°, so any fraction of π is that same fraction of 180. Half of π is 90°, a third of π is 60°.
Example 3
Common mistake: treating an angle in radians as an angle in degrees on the calculator. Computing sin(2) in degree mode gives 0.035, but sin(2 radians) is 0.909 — radically different answers to the same keystrokes. A bare number with no degree symbol means RADIANS, and 2 radians is about 115°. Check the calculator mode before trusting any trig value.
Practice questions
- Convert 90° to radians.
- Convert π/4 radians to degrees.
- What arc length does an angle of π/2 subtend on the unit circle?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Medida en Radianes de Ángulos
Un radián es el ángulo que corta un arco de longitud igual al radio. En el círculo unitario una vuelta completa es 2π radianes, así que 180° equivale a π radianes. Convierte grados a radianes multiplicando por π/180.
Ejemplo 1: Convierte 60° a radianes. Multiplica por π/180: 60 × π/180 = 60π/180 = π/3 radianes. Así que un ángulo de 60° subtiende un arco de longitud π/3 en el círculo unitario.
Ejemplo 2: Convierte 3π/4 radianes a grados. Multiplica por 180/π: (3π/4)(180/π) = 3(180)/4 = 135°. Un anclaje útil: π radianes es media vuelta, 180°, así que cualquier fracción de π es esa misma fracción de 180. La mitad de π es 90°, un tercio de π es 60°.
Ejemplo 3: Error común: tratar un ángulo en radianes como si fuera en grados en la calculadora. Calcular sen(2) en modo grados da 0.035, pero sen(2 radianes) da 0.909 — respuestas radicalmente distintas con las mismas teclas. Un número sin el símbolo de grado significa RADIANES, y 2 radianes son unos 115°. Revisa el modo de la calculadora antes de confiar en cualquier valor trigonométrico.
More Functions lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required