Question Video: Finding the Length of the Arc given Its Central Angle’s Measure and Its Circle’s Radius | Nagwa Question Video: Finding the Length of the Arc given Its Central Angle’s Measure and Its Circle’s Radius | Nagwa

Question Video: Finding the Length of the Arc given Its Central Angle’s Measure and Its Circle’s Radius Mathematics • Third Year of Preparatory School

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Determine to the nearest hundredth, the length of the arc 𝑍𝑌.

02:37

Video Transcript

Determine to the nearest hundredth the length of the arc 𝑍𝑌.

We recall first that this notation means the minor arc connecting points 𝑍 and 𝑌. That’s this arc here. We’ve been asked to calculate the length of this arc. We can recall the formula for doing so. If the central angle of an arc in a circle of radius 𝑟 is 𝜃 degrees, then the arc length 𝑙 is given by 𝑙 equals 𝜃 over 360 multiplied by two 𝜋𝑟. What this formula is doing is calculating the circumference of the full circle, that’s two 𝜋𝑟, and then multiplying this by the portion of the circumference we have. If the central angle is 𝜃, then the fraction of the circumference that the arc represents is 𝜃 over 360, as there are 360 degrees in a full turn.

In this circle, the central angle for the arc 𝑍𝑌, that’s the angle between the two radii connecting the endpoints of this arc to the center of the circle, is 60 degrees. The line segment 𝑀𝑍, which is the diameter of the circle, is of length four centimeters. If the diameter is four centimeters, then the radius is half of this. So it is two centimeters. So, substituting 60 for the central angle 𝜃 and two for the radius 𝑟, we have that the length of the arc 𝑍𝑌 is 60 over 360 multiplied by two multiplied by 𝜋 multiplied by two. 60 over 360 can be simplified to one-sixth by dividing both the numerator and denominator by 60, and two times 𝜋 times two is four 𝜋. So we have one-sixth multiplied by four 𝜋. That’s four 𝜋 over six, which simplifies to two 𝜋 over three.

If we wanted an exact answer or if we didn’t have a calculator, we could leave our answer in this form. But we’ve been asked to determine the length to the nearest hundredth. Evaluating this on a calculator gives 2.0943 continuing. And then rounding to the nearest hundredth or two decimal places gives 2.09. The units are centimeters because we’re finding a length and those were the units given for the diameter in the question. So we found that the length of the arc 𝑍𝑌 to the nearest hundredth is 2.09 centimeters.

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