A wind turbine blade is designed as a sector of a circle with radius 15 meters and central angle \(120^\circ\). Find the perimeter of the blade, including the two straight edges and the curved arc.
["SEO-Optimized Article: Calculating the Perimeter of a Wind Turbine Blade with Circular Sector Geometry", "Wind turbine efficiency relies heavily on blade design, and understanding its geometry is key for engineers and enthusiasts alike. One important aspect is the perimeter of a turbine blade modeled as a circular sector. This article explores how to compute the full perimeter of a turbine blade shaped as a 120° sector with a radius of 15 meters, combining straight radial edges and a curved arc.", "---", "### Introduction to Circular Sector Blade Design", "Wind turbine blades are often designed using curved aerodynamic profiles shaped as portions of circles — known as circular sectors. For performance analysis, calculating the blade’s perimeter is essential, especially when assessing structural dimensions, material usage, or manufacturing specifications. A typical turbine blade ahead of the hub is represented as a sector of a circle with radius ( r = 15 ) meters and central angle ( \ heta = 120^\circ ).", "This geometry consists of two straight edges — the blade’s leading and trailing boundaries — and one curved arc along the outer radius. Knowing the total perimeter helps in design validation, structural analysis, and blade optimization.", "---", "### Step 1: Convert Central Angle to Radians", "Although degrees are intuitive, trigonometric calculations use radians. Convert (120^\circ) to radians:", "[\n\ heta = 120^\circ \ imes \frac{\pi}{180^\circ} = \frac{2\pi}{3} \ ext{ radians}\n]", "---", "### Step 2: Length of the Curved Arc", "The curved part of the blade’s perimeter is the arc length ( s ) of a circle with radius ( r = 15 ) meters and central angle ( \ heta = \frac{2\pi}{3} ) radians. The arc length formula is:", "[\ns = r \ heta\n]", "Substitute the values:", "[\ns = 15 \ imes \frac{2\pi}{3} = 10\pi \approx 31.42 \ ext{ meters}\n]", "---", "### Step 3: Lengths of the Two Straight Radial Edges", "The blade feature two straight edges extending from the center (hub) to the outer tip: both have length equal to the radius ( r = 15 ) meters.", "Total straight edge length:", "[\nL_{\ ext{straight}} = 15 + 15 = 30 \ ext{ meters}\n]", "---", "### Step 4: Total Perimeter of the Blade", "Add the curved arc and the two radial edges:", "[\n\ ext{Perimeter} = s + L_{\ ext{straight}} = 10\pi + 30\n]", "Using ( \pi \approx 3.1416 ), the approximate value is:", "[\n10\pi + 30 \approx 31.42 + 30 = 61.42 \ ext{ meters}\n]", "---", "### Conclusion: Final Blade Perimeter", "The complete perimeter of the wind turbine blade modeled as a 120° circular sector with radius 15 meters is:", "[\n\boxed{10\pi + 30 \ ext{ meters} \approx 61.42 \ ext{ meters}}\n]", "This precise calculation supports engineering design, manufacturing accuracy, and performance modeling in wind energy applications.", "---", "Keywords: wind turbine blade, circular sector, perimeter calculation, 15 meter radius, 120 degree sector, aerodynamic blade design, sector arc length, turbine engineering, renewable energy geometry.", "Page URL:\nHow to Calculate the Perimeter of a Wind Turbine Blade: A 120° Sector with 15m Radius", "Meta Description:\nLearn how to calculate the perimeter of a wind turbine blade shaped as a 120° circular sector with a 15-meter radius. Includes full formula and step-by-step solution for engineers and enthusiasts."]
