The perimeter of the blade includes two radii and the arc length. The arc subtends \(120^\circ\), which is \(\frac{120}{360} = \frac{1}{3}\) of a full circle.

["Understanding the Perimeter of a Blade Shape: Radii, Arc Length, and 120° Subtended Arc", "When analyzing blade geometry—common in engineering, design, and mathematics—the perimeter plays a crucial role in calculating dimensions, material requirements, and aerodynamic or structural performance. One fundamental aspect of blade perimeter calculation involves combining two straight edges (radii) and a curved arc subtending a central angle. This article explains how the perimeter of a blade segment is determined, emphasizing the contribution of a (120^\circ) arc, which represents exactly one-third of a full circle.", "### Blade Perimeter: Radii and Arc Length Combined", "The total perimeter (P) of a blade segment shaped like a circular segment consists of two identical straight-line radii and the curved arc that forms the outer boundary. This hybrid geometry makes the total perimeter:\n[\nP = 2r + L\n]\nwhere (r) is the radius of the circle and (L) is the length of the arc encompassing the blade’s curvature.", "### Arc Length: The Angle Matters", "To calculate arc length (L), engineers use the formula:\n[\nL = r \cdot \ heta\n]\nbut only when (\ heta) is expressed in radians. Since angles are often introduced in degrees, conversion to radians is essential. Recall that:\n[\n\ heta_{\ ext{radians}} = \frac{120^\circ}{360^\circ} \ imes 2\pi = \frac{1}{3} \ imes 2\pi = \frac{2\pi}{3} \ ext{ radians}\n]\nHowever, in the context given, the problem specifies the arc subtends (120^\circ), equivalent to (\frac{120}{360} = \frac{1}{3}) of the full circumference. This reveals that for a (120^\circ) arc on a circle:\n[\nL = \frac{1}{3} \ imes 2\pi r = \frac{2\pi r}{3}\n]", "### Total Perimeter Breakdown", "With the radius (r) known, substitute into the perimeter formula:\n[\nP = 2r + \frac{2\pi r}{3}\n]\nFactoring (r) yields:\n[\nP = r \left(2 + \frac{2\pi}{3}\right)\n]\nThis formula allows engineers and designers to precisely determine blade dimensions using angular measurements, particularly useful in turbine blades, airfoils, and curvilinear tools.", "### Why the (120^\circ) Subtended Arc Matters", "A central angle of (120^\circ) is not arbitrary—it reflects a key angular proportion dividing the circle into three equal parts. This symmetry often appears in optimized designs that balance strength, weight, and flow dynamics. The arc length computes directly from this fraction of the full circumference, ensuring accuracy in manufacturing and performance modeling.", "### Summary", "- The blade perimeter includes two radii and an arc defined by central angle.\n- The subtended arc subtending (120^\circ) corresponds to (\frac{1}{3}) of the full circle.\n- Therefore, arc length is (L = \frac{2\pi r}{3}).\n- The total perimeter combines straight and curved segments: (P = 2r + \frac{2\pi r}{3}).", "Understanding this relationship enables precise design and evaluation of blade geometry, crucial in industries ranging from aviation to renewable energy.", "---", "Mastering the perimeter of geometric blades supports better modeling, efficient material use, and high-performance structural design."]