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: 8 feetCentral Angle, θ: θ = 225°

Verified step by step guidance

1. Convert the central angle from degrees to radians using the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
2. Substitute \( \theta = 225^\circ \) into the formula to find the angle in radians.
3. Use the formula for arc length: \( s = r \times \theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians.
4. Substitute \( r = 8 \) feet and the calculated \( \theta \) in radians into the arc length formula.
5. Simplify the expression to find the arc length in terms of \( \pi \).

Key Concepts

Arc Length Formula

The arc length of a circle can be calculated using the formula L = rθ, where L is the arc length, r is the radius, and θ is the central angle in radians. To use this formula, it's essential to convert the angle from degrees to radians, as the standard unit for angular measurement in trigonometry is radians.

Conversion from Degrees to Radians

To convert an angle from degrees to radians, you can use the conversion factor π radians = 180 degrees. Therefore, to convert degrees to radians, multiply the degree measure by π/180. For example, 225° can be converted to radians by calculating 225 × (π/180), which simplifies to 5π/4 radians.

Expressing in Terms of π

When asked to express the arc length in terms of π, it means to leave the answer in a form that includes π rather than calculating a decimal approximation. This is important in trigonometry as it maintains the exactness of the answer, which can be particularly useful in further calculations or when precise values are required.