STEP 1: find base area $ AB $

To find base area $ AB $ use formula:

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

After substituting $r = 4\, \text{cm}$ we have:

$$ AB = \left( 4\, \text{cm} \right)^{2} \cdot \pi $$ $$ AB = 16\, \text{cm}^2 \cdot \pi $$

STEP 2: find side $ l $

To find side $ l $ use Pythagorean Theorem:

$$ r^2 + h^2 = l^2 $$

After substituting $r = 4\, \text{cm}$ and $h = 2\, \text{cm}$ we have:

$$ \left( 4\, \text{cm} \right)^{2} + \left( 2\, \text{cm} \right)^{2} = l^2 $$ $$ 16\, \text{cm}^2 + 4\, \text{cm}^2 = l^2 $$ $$ l^2 = 20\, \text{cm}^2 $$ $$ l = \sqrt{ 20\, \text{cm}^2 } $$$$ l = 2 \sqrt{ 5 }\, \text{cm} $$

STEP 3: find Curved Surface Area $ CSA $

To find Curved Surface Area $ CSA $ use formula:

$$ CSA = l \cdot r \cdot \pi$$

After substituting $r = 4\, \text{cm}$ and $l = 2 \sqrt{ 5 }\, \text{cm}$ we have:

$$ CSA = 2 \sqrt{ 5 }\, \text{cm} \cdot 4\, \text{cm} \cdot \pi$$$$ CSA = 2 \sqrt{ 5 }\, \text{cm} \cdot 4\, \text{cm} \cdot \pi $$$$ CSA = 8 \sqrt{ 5 }\pi\, \text{cm}^2 $$

STEP 4: find area $ A $

To find area $ A $ use formula:

$$ A = AB + CSA $$

After substituting $AB = 16\pi\, \text{cm}^2$ and $CSA = 8 \sqrt{ 5 }\pi\, \text{cm}^2$ we have:

$$ A = 16\pi\, \text{cm}^2 + 8 \sqrt{ 5 }\pi\, \text{cm}^2 $$ $$ A = 16\pi\, \text{cm}^2 + 8 \sqrt{ 5 }\pi\, \text{cm}^2 $$ $$ A = 106.464\, \text{cm}^2 $$

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