Question

Math

Posted 5 months ago

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A circle has a radius of 3 . An arc in this circle has a central angle of $20^{\circ}$.
What is the length of the arc?
Either enter an exact answer in terms of $\pi$ or use 3.14 for $\pi$ and enter your answer as a decimal.
$\square$
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

To find the arc length, we use the formula $s = r\theta$ where $s$ is the arc length, $r$ is the radius, and $\theta$ is the central angle in radians

step 2

Convert the central angle from degrees to radians. Since $180^\circ$ is equivalent to $\pi$ radians, $20^\circ$ is equivalent to $\frac{20}{180}\pi = \frac{\pi}{9}$ radians

step 3

Substitute the radius $r = 3$ and the central angle $\theta = \frac{\pi}{9}$ radians into the arc length formula: $s = 3 \cdot \frac{\pi}{9}$

step 4

Simplify the expression to find the arc length: $s = \frac{3\pi}{9} = \frac{\pi}{3}$

Answer

The length of the arc is $\frac{\pi}{3}$ or approximately 1.0472 when using $3.14$ for $\pi$.

Key Concept

Arc Length in a Circle

Explanation

The arc length of a circle is calculated using the formula $s = r\theta$, where the central angle $\theta$ must be in radians.

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