Question

s denotes the length of the arc of a circle of radius \(r\) subtended by the central angle \(\theta .\) Find the missing quantity. Round answers to three decimal places. \(r=6\) feet, \(\quad \theta=2\) radians, \(\quad s=?\)

Short answer

12.000 feet

Step-by-step solution

- Identify the given values

The given values are the radius of the circle, which is \(r = 6\) feet, and the central angle, which is \(\theta = 2\) radians. The task is to find the length of the arc, denoted by \(s\).

- Recall the formula for the length of the arc

The formula to find the length of the arc (\(s\)) is given by:\[ s = r \cdot \theta \]

- Substitute the given values into the formula

Substitute \(r = 6\) feet and \(\theta = 2\) radians into the formula:\[ s = 6 \cdot 2 \]

- Perform the multiplication

Multiply 6 by 2 to find the length of the arc:\[ s = 12 \text{ feet} \]

- Round the answer to three decimal places

Since the values used are whole numbers and the result is already exact, the rounded answer is also \(s = 12.000\) feet.

Key concepts

Circle Radius

The radius of a circle is the distance from the center of the circle to any point on its perimeter. It's a fundamental concept in geometry and is represented by the letter \( r \). The radius is crucial in various calculations involving circles, such as the circumference, area, and in this case, the arc length.
Having a clear understanding of the radius helps us delve into more complex topics like the one we are exploring.
In our exercise, the radius \( r \) is given as 6 feet. This means every point on the edge of the circle is exactly 6 feet away from the circle's center. With this simple yet vital piece of information, we can compute other measurements, including the arc length.

Central Angle

The central angle is an angle whose vertex is at the center of the circle, and its sides (or arms) extend to the circumference. In our context, it's represented by \( \theta \).
Central angles are crucial as they directly relate to the arc length of a circle. The arc can be thought of as the 'curved edge' created between two points on the circle's perimeter.
In our problem, the central angle \( \theta \) is given as 2 radians. Radians are a way of measuring angles based on the radius of a circle. One full circle is equivalent to \( 2\pi \) radians, which is roughly 6.283 radians. This information will further help us to calculate the arc length.

Arc Length Formula

To find the length of an arc, we use a specific formula: \[ s = r \cdot \theta \]
Here, \( s \) denotes the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians. This formula is derived from the principle that the arc length is a fraction of the entire circle's circumference. This fraction depends on the angle \( \theta \).
Let's apply this formula with our given values. Our radius \( r \) is 6 feet, and our central angle \( \theta \) is 2 radians.
Substituting these into the formula gives us: \[ s = 6 \cdot 2 = 12 \text{ feet} \]
Thus, the length of the arc is 12 feet. Since our problem mentioned rounding, but the result is already a whole number, we actually have 12.000 feet.
Thus, using this simple yet powerful formula, we can determine the arc length accurately once the radius and central angle are known.