if a circle has a central angle of 85 degrees and a radius of 16 millimeters how long is the arc formed by that angle? leave your answer in terms of pi

Answer 1

To find the length of the arc formed by a central angle in a circle, you can use the formula:

$$\text{Arc Length}=\frac{\theta }{360}\times 2\pi r$$

where:

• $\theta$ is the central angle in degrees,
• $r$ is the radius of the circle.

Given that the central angle $\theta =85$ degrees and the radius $r=16$ millimeters, we can substitute these values into the formula:

$$\text{Arc Length}=\frac{85}{360}\times 2\pi \times 16$$

First, simplify $\frac{85}{360}$:

$$\frac{85}{360}=\frac{17}{72}$$

Now, substitute this back into the formula for arc length:

$$\text{Arc Length}=\frac{17}{72}\times 2\pi \times 16$$

Calculating $2\pi \times 16$:

$$2\pi \times 16=32\pi$$

Now, substituting this into the equation gives:

$$\text{Arc Length}=\frac{17}{72}\times 32\pi$$

Now simplify $\frac{17\times 32\pi }{72}$:

$$\text{Arc Length}=\frac{544\pi }{72}$$

Now simplify $\frac{544}{72}$:

Dividing both the numerator and denominator by 8:

$$\frac{544÷8}{72÷8}=\frac{68}{9}$$

Therefore, the length of the arc is:

$$\text{Arc Length}=\frac{68\pi }{9}\text{ millimeters}$$

Thus, the final answer for the length of the arc, in terms of $\pi$, is:

$$\frac{68\pi }{9}\text{ millimeters}$$