An equilateral triangle is inscribed in a circle of radius 4r. Express the area A within the circle but outside the triangle as a function of the length 5x of the side of the triangle.

Question
An equilateral triangle is inscribed in a circle of radius 4r. Express the area A within the circle but outside the triangle as a function of the length 5x of the side of the triangle.

2020-12-23
In equilateral triangle:
$$\displaystyle\Gamma$$ radius $$\displaystyle={s}{i}{d}\frac{{e}}{\sqrt{{3}}}$$
$$\displaystyle{4}\gamma={5}\frac{{x}}{\sqrt{{3}}}$$
$$\displaystyle{4}\gamma={5}\frac{{x}}{{{4}\sqrt{{3}}}}$$
$$\displaystyle{A}=\pi\times\gamma^{{2}}=\pi{\left(\frac{{{5}{x}}}{{{4}\sqrt{{3}}}}\right)}^{{2}}$$
$$\displaystyle{A}=\frac{{{25}{x}^{{2}}\pi}}{{48}}$$
$$\displaystyle{A}=\frac{\sqrt{{3}}}{{4}}\times{\left({s}{i}{d}{e}\right)}^{{2}}=\frac{\sqrt{{3}}}{{4}}\times{\left({5}{x}\right)}^{{2}}=\frac{{{25}\sqrt{{3}}\times{x}^{{2}}}}{{4}}$$
$$\displaystyle{A}=\frac{{{25}\pi\times{x}^{{2}}}}{{48}}–\frac{{{25}\sqrt{{3}}}}{{4}}\times{x}^{{2}}$$

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