Cone A2V

The surface of the cone in the plane is a circular arc with central angle of 126° and area 415 cm².

Calculate the volume of a cone.

Correct result:

V =  881.1 cm³

Solution:

$$\begin{gathered} A=126\cdot {\displaystyle \frac{\pi }{180}}=126\cdot {\displaystyle \frac{3.1416}{180}}\doteq 2.1991\text{ rad } \\ S=415{\text{cm}}^2 \\ S=\pi s^2\cdot A\mathrm{/}(2\pi ) \\ s=\sqrt{2\cdot S\mathrm{/}A}=\sqrt{2\cdot 415\mathrm{/}2.1991}\doteq 19.4274\text{ cm } \\ r=A\cdot s\mathrm{/}(2\pi )=2.1991\cdot 19.4274\mathrm{/}(2\cdot 3.1416)\doteq 6.7996\text{ cm } \\ h=\sqrt{s^2-r^2}=\sqrt{19.4274^2-6.7996^2}\doteq 18.1986\text{ cm } \\ V={\displaystyle \frac{1}{3}}\cdot \pi \cdot r^2\cdot h={\displaystyle \frac{1}{3}}\cdot 3.1416\cdot 6.7996^2\cdot 18.1986=881.1{\text{cm}}^3 \end{gathered}$$

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