In Exercises 71–74, find the length of the arc on a circle of radius r intercepted by a central angle θ. Express arc length in terms of 𝜋. Then round your answer to two decimal places. Radius, r: 12 inches Central Angle, θ: θ = 45°

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Recall the formula for the length of an arc \(s\) intercepted by a central angle \(\theta\) on a circle of radius \(r\): \(s = r \times \theta_{\text{radians}}\)

Since the central angle \(\theta\) is given in degrees, convert it to radians using the conversion factor: \(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\)

Substitute the given values into the conversion formula: \(\theta_{\text{radians}} = 45 \times \frac{\pi}{180}\)

Now substitute the radius \(r = 12\) inches and the radian measure of \(\theta\) into the arc length formula: \(s = 12 \times \left(45 \times \frac{\pi}{180}\right)\)

Simplify the expression to write the arc length in terms of \(\pi\), then calculate the decimal approximation 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 between two points on the circle. It 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 and radius to the length of the arc.

Conversion Between Degrees and Radians

Since the arc length formula requires the central angle in radians, it is essential to convert degrees to radians. The conversion is done by multiplying degrees by π/180. For example, 45° equals 45 × (π/180) = π/4 radians.

Expressing Answers in Terms of π and Decimal Approximation

After calculating the arc length in terms of π, it is often useful to provide a decimal approximation for practical use. This involves substituting π ≈ 3.1416 and rounding the result to the desired decimal places, such as two decimals, to balance precision and readability.