Find the length to three significant digits of each arc intercepted by a central angle in a circle of radius r. See Example 1. r = 1.38 ft , θ = 5π/6 radians

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Recall the formula for the length of an arc intercepted by a central angle in a circle: \(\text{Arc length} = r \times \theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.

Identify the given values: radius \(r = 1.38\) ft and central angle \(\theta = \frac{5\pi}{6}\) radians.

Substitute the given values into the arc length formula: \(\text{Arc length} = 1.38 \times \frac{5\pi}{6}\).

Multiply the radius by the fraction \(\frac{5}{6}\) and then by \(\pi\) to express the arc length in terms of \(\pi\).

Finally, calculate the numerical value of the arc length and round it to three significant digits to get the final answer.

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Arc Length Formula

The arc length of a circle intercepted by a central angle is calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians. This formula directly relates the angle measure to the length of the arc on the circle.

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Finding Missing Side Lengths

Radian Measure

Radians measure angles based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius. Using radians simplifies arc length calculations since the formula s = rθ requires the angle in radians.

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Significant Figures and Precision

When reporting numerical answers, significant figures indicate the precision of the measurement. Rounding the arc length to three significant digits ensures the result reflects the accuracy of the given data and maintains consistency in reporting.

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