Stosunek pol rombu i koła

[R37]

1. Stosunek pola rombu do pola koła wpisanego w ten romb wynosi 8 : \(\displaystyle{ \Pi}\) . Oblicz miare kata ostrego rombu




2. Kąt ostry ma miare 60stopni. Oblicz stosunek pola kola wpisanego w ten romb do pola tego rombu.

[Swodky]

1.
\(\displaystyle{ sin \alpha = \frac{h}{a} \ \ \ \Rightarrow \ \ h=asin \alpha\\
h=2r\ \ \ \Rightarrow \ \ r=\frac{asin \alpha}{2} \\
\frac{8}{\pi}= \frac{a^2sin \alpha }{\pi(\frac{asin \alpha}{2})^2}\\
8\pi(\frac{asin \alpha}{2})^2= \pi a^2sin \alpha \\
2\pi a^2sin^2 \alpha= \pi a^2sin \alpha \\
2sin \alpha=1\\
sin \alpha= \frac{1}{2} \\
\underline{ \alpha =30 ^{\circ}}}\)

2.
\(\displaystyle{ sin 60 ^{\circ} = \frac{h}{a} \ \ \ \Rightarrow \ \ \ \frac{ \sqrt{3} }{2} = \frac{h}{a}\ \ \ \Rightarrow \ \ \ h=\frac{a \sqrt{3} }{2}\\
h=2r\ \ \ \Rightarrow \ \ r=\frac{a \sqrt{3} }{4} \\
\frac{P _{k} }{P _{r} } = \frac{\pi (\frac{a \sqrt{3} }{4})^2 }{a^2sin 60 ^{\circ} } =
\frac{\pi \frac{3a^2}{16} }{a^2\frac{ \sqrt{3} }{2}}=
\frac{3 \pi a^2 \cdot 2}{a^2 \sqrt{3} \cdot 16}=
\frac{3 \pi }{8\sqrt{3}}=\
\underline{ \frac{\sqrt{3} \pi }{8}}}\)