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All courses Algebra II · F-TF.1 40 of 55
Understand radian measure as arc length on the unit circle.

Define one radian with the arc-length ratio

Problem
Use the diagram to define \(\text{one}~\text{radian}\) by relating intercepted arc length \(s\), radius \(r\), and central angle \(θ\).
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

Define radian measure

\[\theta=s/r\]

For a central angle, radian measure theta is the intercepted arc length s divided by the circle radius r. The two lengths must use the same units.

02

Use the one-radian diagram

\[r=1~\text{and}~\text{one}~\text{radian}~->~s=r=1\]

The inspected diagram shows a unit circle. By definition, one radian intercepts an arc whose length equals one radius.

03

Verify the definition numerically

\[\text{Candidate}~\text{analysis}:~\text{The}~\text{arc}~\text{length}~\text{equals}~\text{the}~\text{radius}:~s~=~r~=~1,~\text{so}~\theta~=~s/r~=~1~\text{radian}.\]

Substituting equal unit lengths gives a unitless ratio of 1, interpreted as one radian.

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Another way

  1. Start from s=r theta and set theta=1; this immediately gives s=r.

!

Common mistake

Do not confuse arc length with straight-line chord length. Radian measure uses distance along the circle divided by radius.

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.

Solution walkthrough video