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Radian Measure
and Arc Length

Discover why radians are the natural way to measure angles, how arc length connects to radius, and derive the key formulas for arc length and sector area.

Precalculus

What Is an Angle, Really?

An angle measures the amount of rotation between two rays sharing a common endpoint. You already know degrees: a full rotation is \(360^\circ\). But there is a more natural unit rooted in the geometry of circles.

\(\theta\) center \(r\) arc

Introducing the Radian

A radian is the angle formed when the arc length equals the radius. Wrap the radius along the circle's edge: the angle you get is exactly 1 radian.

\(r\) \(s = r\) \(1 \text{ rad}\) arc length
= radius

Why Do Radians Matter?

Natural Unit
Radians connect angle directly to arc length: no conversion factor needed.
Calculus Ready
Derivative rules like \(\frac{d}{dx}\sin x = \cos x\) only work in radians.
Universal
Physics, engineering, and computer graphics all default to radian measure.

Deriving \(s = r\theta\) via Similarity

Consider two concentric circles with radii \(r_1\) and \(r_2\), both intercepted by the same central angle \(\theta\). By similarity, the ratio of arc length to radius is the same for every circle:

\(\theta\) \(r_1\) \(r_2\) \(s_1\) \(s_2\)
\(\frac{s_1}{r_1} = \frac{s_2}{r_2}\)
\( = \theta \text{ (const.)}\)

The Arc Length Formula

Formula
$$s = r\theta$$
where \(\theta\) is in radians
Variables
\(s\) : arc length
\(r\) : radius
\(\theta\) : central angle (radians)

Deriving the Sector Area

A sector is a "pizza slice" of a circle. Its area is a fraction of the full circle's area, proportional to the angle:

1
Full circle area: \(A_{\text{circle}} = \pi r^2\)
2
Fraction of circle used by angle \(\theta\): \(\frac{\theta}{2\pi}\)
3
Sector area: \(A = \frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2}r^2\theta\)

Degrees vs Radians

Degrees Radians
Full rotation \(360^\circ\) \(2\pi\)
Half rotation \(180^\circ\) \(\pi\)
Right angle \(90^\circ\) \(\frac{\pi}{2}\)
\(60^\circ\) \(60^\circ\) \(\frac{\pi}{3}\)
\(45^\circ\) \(45^\circ\) \(\frac{\pi}{4}\)
Conversion \(\text{deg} = \text{rad} \times \frac{180}{\pi}\) \(\text{rad} = \text{deg} \times \frac{\pi}{180}\)

Step-by-Step Approach

1
Identify what's given: radius, angle, arc length, or area. Note the angle's unit.
2
Convert to radians if the angle is in degrees: multiply by \(\frac{\pi}{180}\).
3
Choose the formula: Arc length \(s = r\theta\) or Sector area \(A = \frac{1}{2}r^2\theta\).
4
Substitute and solve for the unknown. Simplify, leaving \(\pi\) in exact answers when possible.
5
Check units: arc length is in the same unit as the radius; area is in squared units.

Example 1: Finding Arc Length

A circle has radius \(r = 10 \text{ cm}\). Find the arc length intercepted by a central angle of \(\theta = \frac{\pi}{3}\).

\(\frac{\pi}{3}\) \(10 \text{ cm}\) \(s = ?\)

Example 2: Finding the Angle

An arc of length \(s = 15 \text{ m}\) is on a circle with radius \(r = 6 \text{ m}\). Find the central angle in radians and degrees.

1
Use \(\theta = \frac{s}{r} = \frac{15}{6} = \frac{5}{2} = 2.5\) rad
2
Convert: \(2.5 \times \frac{180}{\pi} \approx 143.2^\circ\)

Example 3: Sector Area

Find the area of a sector with radius \(r = 8 \text{ in}\) and central angle \(120^\circ\).

1
Convert to radians: \(\theta = 120 \times \frac{\pi}{180} = \frac{2\pi}{3}\)
2
Apply formula: \(A = \frac{1}{2}r^2\theta = \frac{1}{2}(8)^2 \cdot \frac{2\pi}{3}\)
3
Simplify: \(A = \frac{1}{2} \cdot 64 \cdot \frac{2\pi}{3} = \frac{64\pi}{3} \approx 67.0 \text{ in}^2\)

Example 4: Finding Radius from Arc

A sector has arc length \(s = 9\pi \text{ cm}\) and central angle \(135^\circ\). Find the radius.

1
Convert: \(\theta = 135 \times \frac{\pi}{180} = \frac{3\pi}{4}\)
2
Solve for \(r\): \(s = r\theta \implies r = \frac{s}{\theta} = \frac{9\pi}{\frac{3\pi}{4}}\)
3
Simplify: \(r = 9\pi \cdot \frac{4}{3\pi} = \frac{36}{3} = 12 \text{ cm}\)

Common Mistakes to Avoid

✕ Using Degrees in Formulas
Plugging \(60\) into \(s = r\theta\) instead of \(\frac{\pi}{3}\). The formulas require radians.
✕ Forgetting the \(\frac{1}{2}\)
Writing \(A = r^2\theta\) for sector area. The correct formula is \(A = \frac{1}{2}r^2\theta\).
✕ Mixing Up \(s\) and \(A\)
Arc length (\(s\)) is linear (\(\text{cm}\)). Sector area (\(A\)) is quadratic (\(\text{cm}^2\)). Check your units!

Formula Summary

Quantity Formula Key Requirement
Radian measure \(\theta = \frac{s}{r}\) \(s\) and \(r\) same units
Arc length \(s = r\theta\) \(\theta\) in radians
Sector area \(A = \frac{1}{2}r^2\theta\) \(\theta\) in radians
Deg → Rad \(\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}\) Multiply, don't add
Rad → Deg \(\theta_{\text{deg}} = \theta_{\text{rad}} \cdot \frac{180}{\pi}\) Multiply, don't add

Question 1 of 5

Convert \(225^\circ\) to radians.

Question 2 of 5

A circle has radius \(r = 5\) cm. What is the arc length intercepted by a central angle of \(\frac{2\pi}{5}\) radians?

Question 3 of 5

Find the area of a sector with radius \(r = 6\) m and central angle \(\frac{\pi}{4}\) radians.

Question 4 of 5

An arc of length \(14\) cm subtends a central angle of \(2\) radians. What is the radius?

Question 5 of 5

Why is the radian measure of an angle defined as the ratio \(\frac{s}{r}\) rather than just the arc length \(s\)?