First, find radius: \( C = 2\pi r = 20\pi \implies r = 10 \, \text{cm} \).

["How to Find the Radius of a Circle When the Circumference Is Known", "Understanding the radius of a circle is essential in geometry and real-world applications, from architecture to engineering. A fundamental formula connects the radius of a circle to its circumference:\n[\nC = 2\pi r\n]\nWhen the circumference ( C ) is given, calculating the radius is simple and straightforward.", "### Step 1: Use the Circumference Formula\nSuppose you are told that the circumference ( C = 20\pi ) centimeters. This is a key clue, as the formula to find the radius is ( r = \frac{C}{2\pi} ).", "Substitute ( C = 20\pi ) into the formula:\n[\nr = \frac{20\pi}{2\pi}\n]", "### Step 2: Simplify the Expression\nThe ( \pi ) terms cancel out, leaving:\n[\nr = \frac{20}{2} = 10\n]", "Thus, the radius ( r = 10 , \ ext{cm} ).", "### Why This Formula Works\nThe formula ( C = 2\pi r ) comes from the definition of ( \pi ): the ratio of the circumference of any circle to its diameter. Since diameter ( d = 2r ), we substitute ( d ) into the ratio:\n[\nC = \pi d = \pi (2r) = 2\pi r\n]\nThis confirms why dividing circumference by ( 2\pi ) yields the radius.", "### Real-World Applications\nKnowing how to derive the radius from circumference helps in:\n- Designing Wheels and Gears: Ensuring proper size matches desired circumference.\n- Crafting Circular Targets: Accurately setting boundaries.\n- Architecture and Construction: Calculating materials for arched structures.", "### Final Answer\nIf the circumference is ( C = 20\pi , \ ext{cm} ), then the radius is:\n[\nr = \frac{C}{2\pi} = \frac{20\pi}{2\pi} = 10 , \ ext{cm}\n]", "Mastering this calculation ensures precision in scientific and practical tasks involving circles — a small radius value with a powerful circular formula.", "---", "Keywords: find radius, circumference formula, calculate radius, circle geometry, C = 2πr, geometry tutorial"]