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 = 12.3 cm , θ = 2π/3 radians

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Recall the formula for the length of an arc intercepted by a central angle \( \theta \) in a circle of radius \( r \): \[ \text{Arc length} = r \times \theta \]

Identify the given values from the problem: radius \( r = 12.3 \) cm and central angle \( \theta = \frac{2\pi}{3} \) radians.

Substitute the given values into the arc length formula: \[ \text{Arc length} = 12.3 \times \frac{2\pi}{3} \]

Multiply the radius by the fraction involving \( \pi \) to express the arc length in terms of \( \pi \): \[ \text{Arc length} = 12.3 \times \frac{2}{3} \times \pi \]

To find the numerical value, multiply \( 12.3 \times \frac{2}{3} \) first, then multiply the result by \( \pi \). Finally, round your answer to three significant digits.

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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 curved segment 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 equals 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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Converting between Degrees & Radians

Significant Figures and Precision

When reporting numerical answers, significant figures indicate the precision of the measurement. Here, the arc length should be rounded to three significant digits, ensuring the result reflects the accuracy of the given data.

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