SL 3.4—The circle, arc and area of sector, degrees only

Arc length is $l=\frac{\theta }{360}\times 2\pi r$.

$l=\frac{90}{360}\times 2\pi (4)=\frac{1}{4}\times 8\pi$.

$l=2\pi$ cm.

The exact arc length is $2\pi$ cm, option A.

$4\pi$ results from using the full circumference $2\pi r=8\pi$ then forgetting to multiply by $\frac{1}{4}$ correctly, giving $r\times \pi$ instead.

$\pi$ cm comes from using $\frac{90}{360}\times \pi r$ instead of $2\pi r$, dropping the factor of 2.

$8\pi$ cm is simply the full circumference, forgetting to take the $\frac{90}{360}$ fraction at all.

Only careful application of the fraction and full formula gives $2\pi$ cm.

Sector area is $A=\frac{\theta }{360}\times \pi r^2$.

$A=\frac{60}{360}\times \pi \times 10^2=\frac{1}{6}\times 100\pi$.

$A=\frac{100\pi }{6}=\frac{50\pi }{3}$ cm².

The exact area is $\frac{50\pi }{3}$ cm², option A.

$\frac{100\pi }{3}$ uses $\frac{60}{180}$ instead of $\frac{60}{360}$, doubling the fraction incorrectly.

$\frac{50\pi }{6}$ divides the correct answer by an extra factor of 2, as if using $r$ instead of $r^2$ partially.

$\frac{25\pi }{3}$ results from using $r^2=25$ (halving the radius) instead of $r^2=100$.

Correctly squaring $r=10$ and using the fraction $\frac{60}{360}$ gives $\frac{50\pi }{3}$ cm².