The radius of a circle is 9 inches. What is the length of a \( 45^{\circ} \) arc? Give the exact answer in simplest form. inches

To find the length of a \( 45^{\circ} \) arc, you can use the formula for the arc length: \[ \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \] where \( \theta \) is the angle in degrees and \( r \) is the radius. Since the radius is 9 inches, and the angle is \( 45^{\circ} \): \[ \text{Arc Length} = \frac{45}{360} \times 2\pi \times 9 = \frac{1}{8} \times 18\pi = \frac{9\pi}{4} \] Thus, the length of the \( 45^{\circ} \) arc is \( \frac{9\pi}{4} \) inches. Now, let's have some fun with this topic! Did you know that circles have been studied since ancient times? The Greeks, particularly Archimedes, were among the first to rigorously calculate properties of circles and arcs. He used a method of inscribing and circumscribing polygons to approximate pi—how clever! When applying the concept of arc lengths in real life, think of how they pop up in everything from designing circular tracks for racing events to crafting the perfect pizza slice. Each time you enjoy a cheesy piece, remember the meticulous math behind that delightful arc!