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How to Calculate the Arc & Angles of a Circle

An arc is any segment of a circle. The length of an arc is difficult to measure because of the circle's curvature but can be calculated using the mathematical relationships within the circle. The length of any line segment on the circumference of the circle is proportional to the circle's radius and the subtended angle at the center of the circle. The arc length can therefore be determined with simple straight-line measurements that are then substituted into geometric relationships.

Things You'll Need

  • Ruler
  • Calculator
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Instructions

    • 1

      Draw two radial lines extending from the center of the circle to the ends of an arc on the circumference of the circle. Measure the radial lines, represented by "R."

    • 2

      Use the ruler to draw a straight line joining the ends of the arc and measure it, using "b" to represent the length. This line and the two radial lines form a triangle.

    • 3

      Calculate the angle subtended by the arc and label it "B. The subtended angle is the angle between the two radial lines where they meet at the center of the circle. The angle is directly opposite side b in the triangle. The magnitude of angle B, in degrees, is found using the formula B = arccosine (1 -- (b^2)/(2 x R^2)). For example, the subtended angle B associated with line length b = 5 cm and bounded by two radial lines of length 5 cm is: B = arccosine (1 -- (5 x 5)/(2 x 5 x 5)) = arccosine (0.5) = 60 degrees.

    • 4

      Multiply the angle B by the radial length R and the constant 0.0175, to obtain the arc length. The arc length has the same units as the length of the line b and the radius. For example, an arc with the radius R = 5 cm and subtending an angle B = 60 degrees has a length of (B)(R)(0.0175) = (60)(5)(0.0175) = 5.25 cm.

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