--- title: "Radian Measure and Sectors" description: "Measure an angle by the arc it cuts on a circle of radius 1, so a full turn is 2π. In radians the arc length is rθ and the sector area is ½r²θ, with no fraction of 360 to carry." canonical: https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4 source: https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Radian Measure and Sectors Measure an angle by the arc it cuts on a circle of radius 1, so a full turn is 2π. In radians the arc length is rθ and the sector area is ½r²θ, with no fraction of 360 to carry. Subject: Mathematics · Area: Geometry · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4 ## Ready when they can - Convert between degrees and radians for 30°, 45°, 60° and 180° - Find an arc length and a sector area with the angle given in radians - State sin, cos and tan of π/6, π/4 and π/3 exactly ## Lesson: Measuring angles with the circle itself Last time you found arcs and sectors as fractions of 360. There is a neater way to measure an angle, using the circle itself: one radian is the angle that cuts off an arc exactly one radius long. A full turn holds 2 pi radians. To convert, multiply degrees by pi over 180. To go back, multiply radians by 180 over pi. Four benchmarks are worth memorising. 30 degrees is pi over 6, 45 is pi over 4, 60 is pi over 3, and 180 is pi. At pi over 6, sine is one half and cosine is root 3 over 2. At pi over 4 both are root 2 over 2. At pi over 3, sine is root 3 over 2 and cosine is one half. Tangent is sine over cosine each time. Radians turn circle formulas simple. Arc length is radius times angle, s equals r theta. A sector is the matching slice, with area half r squared theta. Both need radians. Convert degrees first, then calculate. **Example.** A sector has radius 4 cm and angle pi over 2. Halve the square: half of 16 is 8. Times pi over 2 gives 4 pi square cm. Check against the full circle, 16 pi. The slice is smaller, so the answer is sane. **Recap.** Convert to radians, then multiply: r theta for the arc, half r squared theta for the sector. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Exact Values of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/exact-values-of-sine-cosine-and-tangent-mt_SNJEbXYOrR) - [Arc Length, Sector Area and Segments](https://lightmysky.com/learn/mathematics/arc-length-sector-area-and-segments-mt_z9LIwmfKYJ) ## Opens up - [Rotational Kinematics and the Angular Velocity Vector](https://lightmysky.com/learn/science/rotational-kinematics-and-the-angular-velocity-vector-mt_2UqK-zViqf) - [Double Integrals in Polar Coordinates](https://lightmysky.com/learn/mathematics/double-integrals-in-polar-coordinates-mt_93e-KSzQMT) - [Small Angle Approximations](https://lightmysky.com/learn/mathematics/small-angle-approximations-mt_fIsZes5Os3) - [Polar Coordinates and Polar Curves](https://lightmysky.com/learn/mathematics/polar-coordinates-and-polar-curves-mt_R5QMUJvA1v) - [The Unit Circle and the Graphs of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3-) - [The Argand Plane, Modulus and Argument](https://lightmysky.com/learn/mathematics/the-argand-plane-modulus-and-argument-mt_Wq01c5UN9e)