A circle has circumference \( 20\pi \) cm. Find the area of a sector with central angle \( 72^\circ \).
["How to Find the Area of a Sector Given the Circumference and Central Angle", "Understanding the area of a sector in a circle is essential for solving geometry problems efficiently, especially when the central angle and circumference are known. This article explains step-by-step how to determine the area of a sector with a given circumference and central angle, using the example of a circle with circumference ( 20\pi ) cm and a central angle of ( 72^\circ ).", "---", "### Step 1: Understand the Circle’s Circumference Formula", "The circumference ( C ) of a circle is given by:\n[\nC = 2\pi r\n]\nwhere ( r ) is the radius.", "Given ( C = 20\pi ) cm, we solve for ( r ):\n[\n20\pi = 2\pi r\n]\nDivide both sides by ( 2\pi ):\n[\nr = \frac{20\pi}{2\pi} = 10 \ ext{ cm}\n]", "---", "### Step 2: Find the Central Angle in Radians (Optional for Sector Area Formula)", "The sector area formula uses angular measure in radians:\n[\n\ ext{Sector Area} = \frac{1}{2} r^2 \ heta\n]\nwhere ( \ heta ) is the central angle in radians.", "Since we are given ( \ heta = 72^\circ ), convert it to radians:\n[\n72^\circ = 72 \ imes \frac{\pi}{180} = \frac{2\pi}{5} \ ext{ radians}\n]", "---", "### Step 3: Apply the Sector Area Formula", "[\n\ ext{Sector Area} = \frac{1}{2} r^2 \ heta = \frac{1}{2} (10)^2 \left(\frac{2\pi}{5}\right)\n]\n[\n= \frac{1}{2} \cdot 100 \cdot \frac{2\pi}{5}\n]\n[\n= 50 \cdot \frac{2\pi}{5} = \frac{100\pi}{5} = 20\pi \ ext{ cm}^2\n]", "---", "### Why This Works", "The central angle determines the fraction of the full circle ((360^\circ)) that the sector occupies. Since the total circle area is ( \pi r^2 = \pi (10)^2 = 100\pi ) cm², the sector area is:", "[\n\ ext{Sector Area} = \left(\frac{72}{360}\right) \cdot 100\pi = \frac{1}{5} \cdot 100\pi = 20\pi \ ext{ cm}^2\n]", "This matches the formula and confirms the calculation.", "---", "### Final Answer", "The area of the sector with central angle ( 72^\circ ) and circumference ( 20\pi ) cm is:", "[\n\boxed{20\pi \ ext{ cm}^2}\n]", "---", "Pro Tip: Always convert degrees to radians when using the sector area formula involving ( r^2 \ heta ) for accurate results—this avoids common errors in trigonometric unit conversion."]
