Question

What length of arc is subtended by a central angle of \(75^{\circ}\) on a circle \(13.7\) inches in radius?

Short answer

The length of the arc subtended by a central angle of \(75^{\circ}\) on a circle with a 13.7-inch radius is approximately 18.6 inches.

Step-by-step solution

Write down the arc length formula

The formula to find the arc length is: Arc length = (central angle / 360) * 2π * radius

Plug in the values

We have a central angle of 75 degrees and a circle with a radius of 13.7 inches. Plugging these values into the formula, we get: Arc length = (75 / 360) * 2π * 13.7

Simplify the equation

Now, we need to simplify the equation. First, divide 75 by 360: (75 / 360) = 0.208333 The simplified equation is now: Arc length = 0.208333 * 2π * 13.7

Calculate the arc length

Now, we can multiply 0.208333 by 2π and 13.7 to find the arc length: Arc length = 0.208333 * 6.2832 * 13.7 Arc length ≈ 18.6 inches So, the length of the arc subtended by a central angle of 75 degrees on a circle with a radius of 13.7 inches is approximately 18.6 inches.

Key concepts

Central Angle

The central angle of a circle is the angle whose vertex is at the center of the circle, and whose legs extend to intersect the circle at two points, thereby creating an arc. This angle is significant because it directly determines the size of the arc it subtends. In simpler terms, a larger central angle will encompass a larger portion of the circle's edge, leading to a longer arc, while a smaller angle will subtend a shorter arc.

For example, if you have a slice of pizza where the tip of the slice points towards the center of the whole pizza, the angle at that tip would be the central angle. The crust of your slice represents the arc. Imagine if you increased the angle at the tip of your pizza slice without changing the crust's length; your slice would get bigger but with the same amount of crust. That's how the central angle affects the arc length.

Circle Radius

The radius of a circle is a straight line drawn from the center of the circle to any point on its circumference. It is essentially half the diameter and is a crucial measurement as it defines the size of the circle. The length of the radius affects various circle measurements, including the circumference, area, and, relevant to our context, the arc length.

Understanding the radius is essential because it's a constant that connects all points on a circle's edge. It's like knowing your height—you use it to figure out if you can fit in a space, reach something on a shelf, or throw on a sweater. Similarly, the radius helps you to determine how big the circle is, how much space it takes up, and, when it comes to arcs, how long they are.

Arc Length Formula

To calculate the arc length, we employ a formula that ties together the central angle and the radius of the circle. The formula is:
\[ \text{Arc length} = \left(\frac{\text{central angle}}{360}\right) \times 2\pi \times \text{radius} \]

Consider the arc length formula as a recipe to find out how much edge (crust) your slice of the circle (pizza) has, based on how big the angle (slice) is and the size of the circle (pizza). If we describe the circle in terms of pizza, the radius tells us how large the pizza is, the central angle tells us the size of our slice, and the arc length formula gives us the crust's length. In the given exercise, the formula brings into account the specific central angle (75 degrees) and the unique radius (13.7 inches) to calculate the exact length of the arc.