Arc Length, Radians, and Sector Area
Aligned to G-C.B.5 — Common Core State Standards for Mathematics.
What this lesson teaches
Arc length is proportional to the radius: for a central angle, arc = radius times the angle in radians. One radian is the angle whose arc equals the radius, so a full circle is 2*pi radians. The area of a sector is a fraction of the circle's area equal to the fraction of the full angle it covers.
Worked example
Find the arc length and sector area for a 90° angle in a circle of radius 6. Convert 90° to radians: 90 = pi/2. Arc length = radius * angle = 6 * (pi/2) = 3*pi, about 9.42 units. Sector area = (angle/360)*pi*r^2 = (90/360)*pi*36 = (1/4)*36*pi = 9*pi, about 28.27 square units.
Practice questions
- Convert 60° to radians.
- Find the arc length for a central angle of 2 radians in a circle of radius 5.
- Find the area of a sector with central angle 120° in a circle of radius 9.
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
Longitud de arco, radianes y área del sector
La longitud de arco es proporcional al radio: para un ángulo central, arco = radio por el ángulo en radianes. Un radián es el ángulo cuyo arco es igual al radio, así que un círculo completo es 2*pi radianes. El área de un sector es una fracción del área del círculo igual a la fracción del ángulo total que cubre.
Ejemplo: Halla la longitud de arco y el área del sector para un ángulo de 90° en un círculo de radio 6. Convierte 90° a radianes: 90 = pi/2. Longitud de arco = radio * ángulo = 6 * (pi/2) = 3*pi, unos 9.42 unidades. Área del sector = (ángulo/360)*pi*r^2 = (90/360)*pi*36 = (1/4)*36*pi = 9*pi, unos 28.27 unidades cuadradas.
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