A 45-degree angle is formed by a line and the positive x-axis. If the line intersects a circle of radius 10 cm, what is the length of the arc subtended by this angle?
["Understanding the Arc Length Subtended by a 45-Degree Angle on a Circle", "When visualizing geometric relationships on a circle, one key concept is the arc length subtended by a central angle. In this article, we explore what a 45-degree angle formed by a line and the positive x-axis produces when intersecting a circle—and how to calculate the arc length subtended by that angle on a circle of radius 10 cm.", "---", "### The Basics: Angle, Radius, and Arc Length", "A 45-degree angle at the center of a circle cuts out a special fraction of the circle’s full circumference. Because the full circle spans 360 degrees, the fraction of the circle subtended by a 45-degree angle is:", "$$\n\frac{45^\circ}{360^\circ} = \frac{1}{8}\n$$", "Since the full circumference of a circle is given by:", "$$\nC = 2\pi r\n$$", "the arc length ( L ) corresponding to a central angle ( \ heta ) (in degrees) is:", "$$\nL = \frac{\ heta}{360^\circ} \ imes 2\pi r\n$$", "---", "### Applying the Formula", "Given:\n- Radius ( r = 10,\ ext{cm} )\n- Central angle ( \ heta = 45^\circ )", "Plug values into the formula:", "$$\nL = \frac{45}{360} \ imes 2\pi \ imes 10 = \frac{1}{8} \ imes 20\pi = \frac{20\pi}{8} = \frac{5\pi}{2} ,\ ext{cm}\n$$", "So, the arc length is:", "$$\nL = 2.5\pi ,\ ext{cm} \approx 7.854,\ ext{cm}\n$$", "---", "### Why Is This Important?", "Understanding arc length helps in various real-world applications, from engineering and architecture to physics and computer graphics. Knowing that a 45-degree angle spans one-eighth of a circle makes it easier to calculate distances along curved tracks, design circular components, or analyze rotational motion.", "---", "### Summary", "- A 45-degree angle subtends an 1/8 portion of the circle.\n- With radius 10 cm, the arc length is ( \frac{5\pi}{2} ) cm.\n- This ratio applies universally for circular arcs and simplifies many geometric calculations.", "---", "Key takeaway: Use the central angle fraction multiplied by the circumference to find arc length—especially effective for common angles like 45°, 60°, or 90°. For a 10 cm radius circle, the arc length subtended by a 45° angle is ( \frac{5\pi}{2} ) cm."]
