Convert angle measure and use the radian arc relation

One radian

An angle of one radian subtends an arc equal in length to the radius. In general, the angle is the ratio of arc length to radius, with both lengths in the same units. The ratio is dimensionless, although rad identifies the angle unit. A full turn has measure 2π radians

An arc has length 15 cm on a circle of radius 6 cm. Find its angle in radians

Write 135° as a multiple of π radians. Enter the coefficient of π

Convert 7π/6 radians to degrees

Specify the intended arc

For one circle, a chord other than a diameter has a minor central angle θ in (0,π) and a complementary major angle 2π−θ. Identify which arc is wanted. Arc length is r times the chosen angle in radians; rotation through more than one turn is a separate accumulated-distance question

A circle of radius 8 cm has minor central angle 3π/4. Give the coefficient of π in the major arc length

Which substitution gives the arc length when r=12 cm and θ=30°?

Derive the arc relation, ungraded

Explain why arc length is proportional to radius for a fixed central angle, and why defining θ=l/r gives l=rθ. Use a full turn to justify 360°=2π radians. Independent explanation is ungraded

Model: use similarity and one full turn

Circles are similar, so corresponding arc lengths scale by the same factor as radii; l/r therefore depends only on the angle. A complete circumference is 2πr, giving angle (2πr)/r=2π. Rearranging θ=l/r yields l=rθ. A degree numeral cannot be substituted unchanged because it uses a different unit