# Height as diameter of base

The rotary cylinder has a height equal to the base diameter and the surface of 471 cm2. Calculate the volume of a cylinder.

Correct result:

V =  784.801 cm3

#### Solution:

$S=471 \ \text{cm}^2 \ \\ h=D \ \\ h=2r \ \\ \ \\ S=2 \ \pi r^2 + 2 \ \pi r h \ \\ S=2 \ \pi r^2 + 4 \ \pi r^2 \ \\ S=6 \ \pi r^2 \ \\ \ \\ r=\sqrt{ S/(6 \pi) }=\sqrt{ 471/(6 \cdot \ 3.1416) } \doteq 4.9987 \ \text{cm} \ \\ h=2 \cdot \ r=2 \cdot \ 4.9987 \doteq 9.9975 \ \text{cm} \ \\ \ \\ S_{1}=\pi \cdot \ r^2=3.1416 \cdot \ 4.9987^2=\dfrac{ 157 }{ 2 }=78.5 \ \text{cm}^2 \ \\ \ \\ S_{2}=2 \cdot \ S_{1} + 2 \pi \cdot \ r \cdot \ h=2 \cdot \ 78.5 + 2 \cdot \ 3.1416 \cdot \ 4.9987 \cdot \ 9.9975=471 \ \text{cm}^2 \ \\ \ \\ S_{2}=S \ \\ \ \\ V=S_{1} \cdot \ h=78.5 \cdot \ 9.9975=784.801 \ \text{cm}^3$

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