STEP 1: find side $ a $

To find side $ a $ use formula:

$$ r = \dfrac{ a }{ 2 } $$

After substituting $r = 5\, \text{cm}$ we have:

$$ 5\, \text{cm} = \dfrac{ a }{ 2 } $$ $$ a = 5\, \text{cm} \cdot 2 $$ $$ a = 10\, \text{cm} $$

STEP 2: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $a = 10\, \text{cm}$ we have:

$$ B = \left( 10\, \text{cm} \right)^{2} $$ $$ B = 100\, \text{cm}^2 $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

$$ A = B + L $$

After substituting $A = 125\, \text{cm}$ and $B = 100\, \text{cm}^2$ we have:

$$ 125\, \text{cm} = 100\, \text{cm}^2 + L $$ $$ L = 125\, \text{cm} - 100\, \text{cm}^2 $$ $$ L = 25\, \text{cm} $$

STEP 4: find slant height $ s $

To find slant height $ s $ use formula:

$$ L = 2 \cdot a \cdot s $$

After substituting $L = 25\, \text{cm}$ and $a = 10\, \text{cm}$ we have:

$$ 25\, \text{cm} = 2 \cdot 10\, \text{cm} \cdot s $$$$ 25\, \text{cm} = 20\, \text{cm} \cdot s $$$$ s = \dfrac{ 25\, \text{cm} }{ 20\, \text{cm} } $$$$ s = \frac{ 5 }{ 4 } $$

STEP 5: find height $ h $

To find height $ h $ use Pythagorean Theorem:

$$ h^2 + \frac{ a^2 }{ 4 } = s^2 $$

After substituting $a = 10\, \text{cm}$ and $s = \dfrac{ 5 }{ 4 }\, \text{cm}^0$ we have:

$$ h ^ {\,2} + \frac{ \left( 10\, \text{cm} \right)^{2} }{ 4 } = \frac{ 5 }{ 4 } $$ $$ h ^ {\,2} + \frac{ 100\, \text{cm}^2 }{ 4 } = \frac{ 5 }{ 4 } $$ $$ h ^ {\,2} + 25\, \text{cm}^2 = \frac{ 5 }{ 4 } $$ $$ h ^ {\,2} = \frac{ 25 }{ 16 } - 25\, \text{cm}^2 $$ $$ h ^ {\,2} = -\frac{ 375 }{ 16 } $$

This equation has no solution $ \Longrightarrow $ The problem has no solution.

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