Find the length of the arc on a circle of radius 20 feet intercepted by a 75° central angle. Express arc length in terms of 𝜋. Then round your answer to two decimal places.

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Identify the given values: the radius \(r = 20\) feet and the central angle \(\theta = 75^\circ\).

Recall the formula for the length of an arc: \(L = r \times \theta\), where \(\theta\) must be in radians.

Convert the central angle from degrees to radians using the conversion \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\), so calculate \(75^\circ \times \frac{\pi}{180}\).

Substitute the radius and the radian measure of the angle into the arc length formula: \(L = 20 \times \left(75 \times \frac{\pi}{180}\right)\).

Simplify the expression to write the arc length in terms of \(\pi\), then use a calculator to approximate the decimal value and round it to two decimal places.

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주요 개념

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Arc Length Formula

The arc length of a circle is the distance along the curved line forming part of the circle's circumference. It is calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians. This formula connects linear and angular measurements on a circle.

Conversion Between Degrees and Radians

Angles can be measured in degrees or radians. Since the arc length formula requires the angle in radians, convert degrees to radians by multiplying by π/180. For example, 75° equals (75 × π/180) radians, which is essential for accurate calculation.

Rounding and Expressing Answers in Terms of π

Expressing the arc length in terms of π keeps the answer exact and symbolic. After finding the exact value, you can approximate the decimal value by substituting π ≈ 3.1416 and rounding to the desired decimal places, such as two decimals, for practical use.