Find the area $ A $ of a cone if radius $r = 20\, \text{cm}$ and height $h = 99\, \text{cm}$.

$2420\pi\, \text{cm}^2$

STEP 1: find base area $ AB $

To find base area $ AB $ use formula:

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

After substituting $r = 20\, \text{cm}$ we have:

$$ AB = \left( 20\, \text{cm} \right)^{2} \cdot \pi $$ $$ AB = 400\, \text{cm}^2 \cdot \pi $$

STEP 2: find side $ l $

To find side $ l $ use Pythagorean Theorem:

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

After substituting $r = 20\, \text{cm}$ and $h = 99\, \text{cm}$ we have:

$$ \left( 20\, \text{cm} \right)^{2} + \left( 99\, \text{cm} \right)^{2} = l^2 $$ $$ 400\, \text{cm}^2 + 9801\, \text{cm}^2 = l^2 $$ $$ l^2 = 10201\, \text{cm}^2 $$ $$ l = \sqrt{ 10201\, \text{cm}^2 } $$$$ l = 101\, \text{cm} $$

STEP 3: find Curved Surface Area $ CSA $

To find Curved Surface Area $ CSA $ use formula:

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

After substituting $r = 20\, \text{cm}$ and $l = 101\, \text{cm}$ we have:

$$ CSA = 101\, \text{cm} \cdot 20\, \text{cm} \cdot \pi$$$$ CSA = 101\, \text{cm} \cdot 20\, \text{cm} \cdot \pi $$$$ CSA = 2020\pi\, \text{cm}^2 $$

STEP 4: find area $ A $

To find area $ A $ use formula:

$$ A = AB + CSA $$

After substituting $AB = 400\pi\, \text{cm}^2$ and $CSA = 2020\pi\, \text{cm}^2$ we have:

$$ A = 400\pi\, \text{cm}^2 + 2020\pi\, \text{cm}^2 $$ $$ A = 400\pi\, \text{cm}^2 + 2020\pi\, \text{cm}^2 $$ $$ A = 2420\pi\, \text{cm}^2 $$

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