Circular segment

Calculate the radius and area of the circular segment if the center angle = 106° and the length of the corresponding circular arc is l = 52 cm.

Final Answer:

r =  28.1074 cm
S =  730.7915 cm²

Step-by-step explanation:

$$\begin{gathered} \alpha =106{}^{\circ } \\ l=52\text{cm} \\ k=\frac{\alpha }{360}=\frac{106}{360}=\frac{53}{180}\doteq 0.2944 \\ o=2\pi r \\ l=k\cdot 2\pi \cdot r \\ r=\frac{l}{2\pi \cdot k}=\frac{52}{2\cdot 3.1416\cdot 0.2944}\doteq 28.1074\text{cm} \\ \text{ Verifying Solution: } \\ o=2\pi \cdot r=2\cdot 3.1416\cdot 28.1074=\frac{9360}{53}\doteq 176.6038\text{cm} \end{gathered}$$

$$\begin{gathered} S_1=\pi \cdot r^2=3.1416\cdot 28.1074^2\doteq 2481.9332{\text{cm}}^2 \\ S=k\cdot S_1=0.2944\cdot 2481.9332=730.7915{\text{cm}}^2 \end{gathered}$$

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