Circles Pattern - Arcs Angles Radians

This pattern covers arc lengths, central angles, and conversions between degrees and radians.

 

Degree–Radian Conversion

To convert degrees to radians: multiply by $\dfrac{\pi}{180}$. To convert radians to degrees: multiply by $\dfrac{180}{\pi}$.

What is $150°$ in radians?

A) $\dfrac{3\pi}{5}$
B) $\dfrac{5\pi}{6}$
C) $\dfrac{5\pi}{3}$
D) $\dfrac{3\pi}{2}$

$150 \times \dfrac{\pi}{180} = \dfrac{150\pi}{180} = \dfrac{5\pi}{6}$. The answer is B.

In the circle with center $C$, the measure of $\angle ACB$ is $\dfrac{2\pi}{5}$ radians. What is the measure of minor arc $AB$ in degrees?

The arc measure equals the central angle. Convert: $\dfrac{2\pi}{5} \times \dfrac{180}{\pi} = \dfrac{360}{5} = 72°$.

 

Arc Length and Circumference

The arc length is proportional to the central angle:

$\dfrac{\text{arc length}}{\text{circumference}} = \dfrac{\text{central angle}}{360°}$

So: arc length $= \dfrac{\theta}{360} \times 2\pi r$ (if $\theta$ is in degrees), or arc length $= r\theta$ (if $\theta$ is in radians).

A circle with center $P$ has arc $QR$ of length $15\pi$ cm, with central angle $120°$. What is the circumference?

A) $15\pi$
B) $45\pi$
C) $5\pi$
D) $30\pi$

$\dfrac{15\pi}{C} = \dfrac{120}{360} = \dfrac{1}{3}$. So $C = 3 \times 15\pi = 45\pi$. The answer is B.

An arc of $60°$ has length 7 meters. What is the full circumference?

$\dfrac{7}{C} = \dfrac{60}{360} = \dfrac{1}{6}$. So $C = 42$ meters.

A circular stained-glass window has a minor arc of $72°$ with length $5\pi$ cm. What is the radius?

A) $5$
B) $10$
C) $25$
D) $12.5$

Arc length $= \dfrac{72}{360} \times 2\pi r = \dfrac{1}{5} \times 2\pi r = \dfrac{2\pi r}{5}$. Set equal to $5\pi$: $\dfrac{2\pi r}{5} = 5\pi$, so $r = \dfrac{25}{2} = 12.5$. The answer is D.

 

Arc Measure Equals Central Angle

The degree measure of an arc equals the degree measure of its central angle. This is a definition, not a formula — just convert if needed.

The length of minor arc $AB$ is $\dfrac{5}{12}$ of the circumference. What is the central angle $\angle APB$?

If the arc is $\dfrac{5}{12}$ of the circumference, the central angle is $\dfrac{5}{12} \times 360° = 150°$.

 

Two Diameters: Arc Relationships

When two diameters cross at the center, they create four arcs. Vertical angles are equal, so opposite arcs are equal.

A circle has center $K$ and circumference $90\pi$. Diameters $\overline{AC}$ and $\overline{BD}$ cross at $K$. The arc $AB$ is four times the arc $BC$. What is the length of arc $AB$?

A) $9\pi$
B) $36\pi$
C) $45\pi$
D) $72\pi$

Let arc $BC = x$. Then arc $AB = 4x$. Since $\overline{AC}$ is a diameter, arcs $AB + BC = $ semicircle $= \dfrac{90\pi}{2} = 45\pi$. So $4x + x = 45\pi$, giving $x = 9\pi$ and arc $AB = 36\pi$. The answer is B.

A circle has circumference $180\pi$. Diameters $\overline{AC}$ and $\overline{BD}$ cross at center $M$. Arc $AB$ is $\dfrac{1}{4}$ of arc $BC$. What is arc $BC$?

A) $18\pi$
B) $45\pi$
C) $72\pi$
D) $90\pi$

Let arc $AB = x$, so arc $BC = 4x$. Semicircle: $x + 4x = 90\pi$, giving $x = 18\pi$ and arc $BC = 72\pi$. The answer is C.

 

Evaluating Trig Functions with Large Radian Arguments

When the angle is large (like $\dfrac{91\pi}{3}$), reduce it using the period of the function. Both $\sin$ and $\cos$ have period $2\pi$.

What is $\sin!\left(\dfrac{91\pi}{3}\right)$?

A) $\dfrac{1}{2}$
B) $\dfrac{\sqrt{3}}{2}$
C) $-\dfrac{1}{2}$
D) $-\dfrac{\sqrt{3}}{2}$

Divide $\dfrac{91}{3}$ by $2$ (one full period is $2\pi$): $\dfrac{91}{3} \div 2 = \dfrac{91}{6} = 15\,\text{R}\,\dfrac{1}{6}$. So $\dfrac{91\pi}{3} = 15(2\pi) + \dfrac{\pi}{6} + \dfrac{\pi}{3}$. Actually, simplify: $\dfrac{91}{3} = 30 + \dfrac{1}{3}$. So $\dfrac{91\pi}{3} = 30\pi + \dfrac{\pi}{3}$. Since $30\pi = 15(2\pi)$, the angle reduces to $\dfrac{\pi}{3}$. $\sin!\left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{2}$. The answer is B.

What is $\cos(17\pi)$?

A) $1$
B) $-1$
C) $0$
D) $\dfrac{1}{2}$

$17\pi = 8(2\pi) + \pi$. So $\cos(17\pi) = \cos(\pi) = -1$. The answer is B.

Reduction recipe: Divide the coefficient of $\pi$ by 2. The remainder tells you the equivalent angle. If the remainder is 0, the angle is a multiple of $2\pi$ (same as $0$). If $1$, same as $\pi$.

 

What to Do on Test Day

• Conversions to memorize:
• Degrees to radians: multiply by $\dfrac{\pi}{180}$
• Radians to degrees: multiply by $\dfrac{180}{\pi}$
• Common values: $30° = \dfrac{\pi}{6}$, $45° = \dfrac{\pi}{4}$, $60° = \dfrac{\pi}{3}$, $90° = \dfrac{\pi}{2}$, $180° = \pi$
• Arc length formula: $\text{arc length} = \dfrac{\theta}{360} \times 2\pi r$ (degrees) or $\text{arc length} = r\theta$ (radians).
• Arc fraction shortcut: $\dfrac{\text{arc length}}{\text{circumference}} = \dfrac{\text{central angle}}{360°}$. If you know any two of the three values (arc, circumference, angle), you can find the third.
• Two diameters: Opposite arcs are equal (vertical angles). Adjacent arcs formed by a diameter sum to a semicircle.
• Reducing large angles: To evaluate $\sin$ or $\cos$ of a large angle, divide the coefficient of $\pi$ by 2. The remainder (times $\pi$) gives you the equivalent angle in $[0, 2\pi)$. Then use the unit circle.
• Arc measure = central angle in degrees. This is a definition, not something to derive.