Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). Radius \(r\) 14.5 centimeters Arc Length \(s\) 35 centimeters

Step-by-Step Solution

Answer

The radian measure of the central angle is approximately 2.41 radians.

Step 1: Identify Given Variables

Here, the radius \(r\) is given as 14.5 cm and arc length \(s\) is given as 35 cm.

Step 2: Apply the Radian Measure Formula

Using the radian measure formula \(θ = s/r\), substitute the given values into the formula.

Step 3: Calculation

Upon substitution, we get θ = 35/14.5 = 2.41 radians. Hence, the radian measure of the central angle is approximately 2.41.

Key Concepts

Central Angle

A central angle is an angle whose vertex is at the center of a circle, and whose sides are radii that intersect the circle. It plays a crucial role in the radian measure formula, which links the central angle, the arc length, and the radius of the circle. This angle is typically measured in radians when we are using the formula \[ θ = \frac{s}{r} \]where \(θ\) is the central angle in radians, \(s\) is the arc length, and \(r\) is the radius. Understanding the central angle helps in relating the dimensions of a circle to specific angular measures, making it a key component in circular geometry.

Circle

A circle is a shape with all points at a fixed distance, known as the radius, from a central point called the center. It is a fundamentally important shape in geometry and has properties that are used in various mathematical contexts. When thinking about the circle in relation to central angles, remember:

• A full circle corresponds to an angle of \(2π\) radians.
• The circumference of the circle is \(2πr\), where \(r\) is the radius.
• Each section of the circumference corresponding to a central angle equals the arc length.

These properties help us understand how arc lengths and central angles are interrelated.

Arc Length

Arc length is the measure of the distance along the curved line making up an arc on a circle. When dealing with circles, arcs are portions of the circle's circumference. The arc length can be easily found when you know the radius of the circle and the central angle. By using the formula \[ s = θ \cdot r \] where \(s\) is the arc length, \(θ\) is the central angle in radians, and \(r\) is the radius, you can determine the length of the arc for any given circle. Note that the arc length is essentially the part of the circle's circumference cut off by the central angle, thus closely tying it to our other concepts.

Radius

The radius of a circle is the fixed distance from the center of the circle to any point on its circumference. It is a pivotal measurement as it relates to both the area of the circle and its arc lengths. When using the radian measure formula, the radius is part of the division equation where arc length \(s\) is divided by \(r\) to find the central angle \(θ\)\:

• The radius is crucial in measuring how large the circle is, affecting all properties of the circle including arc measures.
• It is always half of the diameter, which can sometimes be used to derive or confirm it.

In any geometric figure involving circles, being aware of the radius is necessary for calculating precise geometric properties including area, circumference, and angles.