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Petal perimeter: combine equal arcs

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      Geometry · Arc length

      Question

      Each petal in the diagram has the same radius as the small circle. The small circle has circumference 6.28 cm. Find the perimeter of the flower.

      Petal perimeter and arc lengthsFour symmetric petals each have a 270-degree outer arc. Together they equal three complete circumferences.Outer petal arc = ¾ circumferenceTotal: 4 × ¾ × 6.28 = 18.84 cmOne outer arc (270°)
      Each outer petal arc covers 270°, or three quarters of a circle. Four petals total three complete circumferences.

      Step-by-step solution

      1. Identify the outer boundary

      The flower's perimeter consists of four identical outer arcs. Each arc has the same radius as the small circle.

      2. Find the fraction for one arc

      The inner angle at the join is 90°, so the outer arc has central angle 360° − 90° = 270°. Its length is 270°/360° = 3/4 of the small circle's circumference.

      3. Combine the four arcs

      Their total is 4 × 3/4 = 3 full circumferences. There is no need to calculate the radius first.

      4. Substitute the given circumference

      The perimeter is 3 × 6.28 = 18.84 cm.

      The flower's perimeter is 18.84 cm.

      Which method does this problem use?

      Use symmetry and the central-angle fraction to combine equal arcs. Working with the given circumference directly avoids introducing an unnecessary approximation for π.