Equilateral triangle vs circle

Find the area of an equilateral triangle inscribed in a circle of radius r = 9 cm. What percentage of the circle area does it occupy?

Correct answer:

S =  105.2221 cm²
p =  41.3497 %

Step-by-step explanation:

$$\begin{gathered} r=9\text{ cm } \\ a=3\mathrm{/}\sqrt{3}\cdot r=3\mathrm{/}\sqrt{3}\cdot 9=9\sqrt{3}\text{ cm}\doteq 15.5885\text{ cm } \\ S={\displaystyle \frac{\sqrt{3}}{4}}\cdot a^2={\displaystyle \frac{\sqrt{3}}{4}}\cdot 15.5885^2=105.2221{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} S_1=\pi \cdot r^2=3.1416\cdot 9^2\doteq 254.469{\text{cm}}^2 \\ p=100\cdot {\displaystyle \frac{S}{S_1}}=100\cdot {\displaystyle \frac{105.2221}{254.469}}=41.3497\mathrm{\%} \end{gathered}$$

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