STEP 1: find base area $ AB $

To find base area $ AB $ use formula:

$$ AB = r^2 \cdot \pi $$

After substituting $r = 6\, \text{cm}$ we have:

$$ AB = \left( 6\, \text{cm} \right)^{2} \cdot \pi $$ $$ AB = 36\, \text{cm}^2 \cdot \pi $$

STEP 2: find lateral area $ AL $

To find lateral area $ AL $ use formula:

$$ A = 2 AB + AL $$

After substituting $A = 561.432\, \text{cm}$ and $AB = 36\pi\, \text{cm}^2$ we have:

$$ 561.432\, \text{cm} = 2 \cdot 36\pi\, \text{cm}^2 + AL $$ $$ 561.432\, \text{cm} = 72\pi\, \text{cm}^2 + AL $$ $$ AL = 561.432\, \text{cm} - 72\pi\, \text{cm}^2 $$ $$ AL = 335.2373\, \text{cm} $$

STEP 3: find height $ h $

To find height $ h $ use formula:

$$ AL = 2 \cdot h \cdot r \cdot \pi$$

After substituting $AL = 335.2373\, \text{cm}$ and $r = 6\, \text{cm}$ we have:

$$ 335.2373\, \text{cm} = 2 \cdot h \cdot \left( 6\, \text{cm} \right)^{4} \cdot \pi$$$$ 335.2373\, \text{cm} = 12\, \text{cm} \cdot h \cdot \pi $$$$ h = \dfrac{ 335.2373\, \text{cm}}{ 12\, \text{cm} \, \pi } $$$$ h \approx 8.8924 $$

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