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Math  /  Geometry

QuestionGiven: r=32 centimeters r = 32 \text{ centimeters} θ=π4 radians \theta = \frac{\pi}{4} \text{ radians}
Find the length in cm.

Studdy Solution

STEP 1

1. The given problem involves finding the length of an arc of a circle.
2. The length of an arc s s can be found using the formula s=rθ s = r\theta , where r r is the radius and θ \theta is the central angle in radians.
3. The radius r r is given as 32 centimeters.
4. The central angle θ \theta is given as π4 \frac{\pi}{4} radians.

STEP 2

1. Identify the values of r r and θ \theta .
2. Use the formula for the length of an arc s=rθ s = r\theta .
3. Substitute the given values into the formula.
4. Simplify the expression to find the length of the arc in centimeters.

STEP 3

Identify the given values of r r and θ \theta .
Given: r=32 centimeters r = 32 \text{ centimeters} θ=π4 radians \theta = \frac{\pi}{4} \text{ radians}

STEP 4

Recall the formula for the length of an arc s s of a circle with radius r r and central angle θ \theta :
s=rθ s = r\theta

STEP 5

Substitute the given values r=32 r = 32 centimeters and θ=π4 \theta = \frac{\pi}{4} radians into the formula s=rθ s = r\theta .
s=32⋅π4 s = 32 \cdot \frac{\pi}{4}

STEP 6

Simplify the expression to find the length of the arc s s .
s=32⋅π4=8π centimeters s = 32 \cdot \frac{\pi}{4} = 8\pi \text{ centimeters}
Therefore, the length of the arc is 8π 8\pi centimeters.

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