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All courses Math III · F-TF.1 33 of 55
Understand radian measure as arc length on the unit circle.

Define one radian with the arc-length ratio

Problem
For a unit circle, complete the definition of one radian. Enter radius r, intercepted arc length s, the required relation between them, and theta=s/r with its unit.
Prompt circle or wheel diagram with two radii and a directed intercepted arc; only quantities stated in the problem are labeled. Open full size
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Hint

Write theta=s/r before substituting the arc length and radius.

For a central angle measured in radians, theta=s/r. One radian occurs exactly when s=r.

Solution walkthrough

01

Record the radius

\[r=1\]

The radius supplies the reference length used to measure the intercepted arc.

02

Record the arc length

\[s=1\]

For one radian, the intercepted arc has exactly the same length as the radius.

03

Use the radian ratio

\[\text{theta}=s/r\]

Radian measure is dimensionless arc length divided by radius.

04

Evaluate the ratio

\[\text{theta}=1/1=1~\text{radian}\]

Equal numerator and denominator give a central angle of one radian.

05

Submit all fields

\[\text{radius}~r=1;~\text{arc}~\text{length}~s=1;~\text{defining}~\text{relation}=s=r;~\text{angle}~\text{theta}=s/r=1/1=1~\text{radian}\]

The closed fields state the definition without relying on open prose.

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Another way

  1. Starting from s=r theta, substitute theta=1 to obtain s=r.

!

Common mistake

Do not use the diameter, invert s/r, or confuse one radian with one degree.

Answer circle or wheel diagram with the exact directed arc, central angle, radius relation, and requested result labeled.
One radian occurs when the intercepted arc has the same length as the radius.