Find the length of the arc on a circle of radius 10 feet intercepted by a 135° central angle. Express arc length in terms of 𝜋. Then round your answer to two decimal places.

Verified step by step guidance

1. Convert the central angle from degrees to radians using the formula: radians = degrees * (\frac{\pi}{180}).
2. Calculate the radian measure for 135°: 135 * (\frac{\pi}{180}).
3. Use the formula for arc length: arc length = radius * angle in radians.
4. Substitute the radius (10 feet) and the radian measure of the angle 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 degrees to radians by multiplying the degree measure by π/180. This relationship allows for the calculation of the arc length based on the circle's radius and the angle subtended at the center.

Conversion from Degrees to Radians

To apply the arc length formula, angles must be in radians. The conversion from degrees to radians is done using the formula radians = degrees × (π/180). For example, a 135° angle converts to 135 × (π/180) = 3π/4 radians. Understanding this conversion is crucial for accurately calculating the arc length.

Circumference of a Circle

The circumference of a circle is the total distance around it, calculated using the formula C = 2πr, where r is the radius. Knowing the circumference helps in understanding the relationship between the arc length and the total circle. The arc length is a fraction of the circumference, determined by the ratio of the central angle to 360°, which is essential for interpreting the results of the arc length calculation.