Question

the formula for the length of an arc is:
a. ( a = 2pi rleft(\frac{\theta}{360}
ight) )
b. ( a = pi d^{2}left(\frac{\theta}{360}
ight) )
c. ( a = pi rleft(\frac{\theta}{180}
ight) )
d. ( a = pi r^{2}left(\frac{\theta}{360}
ight) )

Explanation:

Step1: Recall the formula for arc length

The formula for the length of an arc \(s\) (here \(A\) represents arc length) is \(s = 2\pi r\times\frac{\theta}{360}\) (where \(r\) is the radius of the circle and \(\theta\) is the central angle in degrees).

Step2: Analyze each option

• Option a: \(A = 2\pi r(\frac{\theta}{360})\) matches the arc - length formula.
• Option b: \(A=\pi d^{2}(\frac{\theta}{360})\), since \(d = 2r\), \(\pi d^{2}\) is related to the area of a circle (\(A=\pi r^{2}\), and if \(d = 2r\), then \(A=\pi(\frac{d}{2})^{2}=\frac{\pi d^{2}}{4}\)), this is not the arc - length formula.
• Option c: \(A=\pi r(\frac{\theta}{180})\) is incorrect. The full - circle circumference is \(C = 2\pi r\), and when we calculate the arc length, we should use the proportion based on \(360^{\circ}\) for a full circle.
• Option d: \(A=\pi r^{2}(\frac{\theta}{360})\) is the formula for the area of a sector, not the arc length.

Answer:

a. \(A = 2\pi r(\frac{\theta}{360})\)