How do you use the arc length formula to find the radius when given the arc length, central angle, and other related measurements?

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The arc length formula is a valuable tool in mathematics used to calculate the length of an arc in a circle. The formula is given by:

Arc Length (L) = r * θ

where:

  • L is the arc length.
  • r is the radius of the circle.
  • θ is the central angle in radians.

In this case, we are provided with the following information:

  • Arc Length (L) = 12 cm
  • Central Angle (θ) = 36°

First, we need to convert the angle from degrees to radians since the formula requires the angle in radians. The conversion from degrees to radians can be done using the formula:

Radians = Degrees × (π / 180)

Applying this to our central angle:

θ = 36° × (π / 180) = π / 5 radians

Now that we have the angle in radians, we can substitute the values back into the arc length formula:

12 cm = r * (π / 5)

To isolate r, rearranging the equation gives us:

r = 12 cm / (π / 5)

This can be simplified further:

r = 12 cm * (5 / π)

Finally, calculating this gives us the radius:

r ≈ 12 cm * 1.59265 ≈ 19.11 cm

So, the radius r of the circle is approximately 19.11 cm.

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