Solomon needs to justify the formula for the arc length of a sector. Which expression best completes this argument?The circumference of a circle is given by the formula , where d is the diameter.Because the diameter is twice the radius, .If equally sized central angles, each with a measure of , are drawn, the number of sectors that are formed will be equal to _____.The arc length of each sector is the circumference divided by the number of sectors, or .Therefore, the arc length of a sector of a circle with a central angle of is given by or .
Question
Solomon needs to justify the formula for the arc length of a sector. Which expression best completes this argument?The circumference of a circle is given by the formula , where d is the diameter.Because the diameter is twice the radius, .If equally sized central angles, each with a measure of , are drawn, the number of sectors that are formed will be equal to _____.The arc length of each sector is the circumference divided by the number of sectors, or .Therefore, the arc length of a sector of a circle with a central angle of is given by or .
Solution
The missing expression in your argument is "360/θ". Here's the completed argument:
The circumference of a circle is given by the formula C = πd, where d is the diameter. Because the diameter is twice the radius, C = 2πr. If equally sized central angles, each with a measure of θ, are drawn, the number of sectors that are formed will be equal to 360/θ. The arc length of each sector is the circumference divided by the number of sectors, or (2πr)/(360/θ). Therefore, the arc length of a sector of a circle with a central angle of θ is given by (θ/360) * 2πr or (θ/180)πr.
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