Radians Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Find the arc length of a sector with radius 1010 cm and central angle 3ฯ€4\frac{3\pi}{4} radians.

Solution

  1. 1
    The arc length formula is s=rฮธs = r\theta, where ฮธ\theta is in radians.
  2. 2
    Substitute: s=10โ‹…3ฯ€4=30ฯ€4=15ฯ€2s = 10 \cdot \frac{3\pi}{4} = \frac{30\pi}{4} = \frac{15\pi}{2}.
  3. 3
    s=15ฯ€2โ‰ˆ23.56s = \frac{15\pi}{2} \approx 23.56 cm.

Answer

s=15ฯ€2โ‰ˆ23.56ย cms = \frac{15\pi}{2} \approx 23.56 \text{ cm}
The formula s=rฮธs = r\theta is one of the main reasons radians are used in mathematics: it gives a simple, direct relationship between arc length, radius, and angle. In degrees, the formula would be s=ฯ€rฮธ180s = \frac{\pi r \theta}{180}, which is less elegant.

About Radians

A radian is an angle measurement defined by the arc length it subtends on a unit circle: one radian is the angle at which the arc length equals the radius. A full circle is 2ฯ€2\pi radians (about 6.28 radians), making radians the natural unit for trigonometry and calculus.

Learn more about Radians โ†’

More Radians Examples

Example 1 easy

Convert [formula] to radians.

Example 3 medium

Convert [formula] radians to degrees and identify which quadrant this angle is in.

Example 4 hard

A wheel of radius [formula] cm rotates at [formula] rpm (revolutions per minute). Find the angular v

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