STEP 1: find side $ a $

To find side $ a $ use formula:

After substituting $B = 6.5\, \text{cm}$ we have:

$$ B = a^2 $$ $$ a^2 = 6.5\, \text{cm} $$ $$ a = \sqrt{ 6.5\, \text{cm} } $$$$ a \approx 2.5495 $$

STEP 2: find lateral surface $ L $

To find lateral surface $ L $ use formula:

$$ A = B + L $$

After substituting $A = 126.42\, \text{cm}$ and $B = 6.5\, \text{cm}$ we have:

$$ 126.42\, \text{cm} = 6.5\, \text{cm} + L $$ $$ L = 126.42\, \text{cm} - 6.5\, \text{cm} $$ $$ L = 119.92\, \text{cm} $$

STEP 3: find slant height $ s $

To find slant height $ s $ use formula:

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

After substituting $L = 119.92\, \text{cm}$ and $a = 2.5495\, \text{cm}^0$ we have:

$$ 119.92\, \text{cm} = 2 \cdot 2.5495 \cdot s $$$$ 119.92\, \text{cm} = 5.099 \cdot s $$$$ s = \dfrac{ 119.92\, \text{cm} }{ 5.099 } $$$$ s = 23.5182\, \text{cm} $$

STEP 4: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $a = 2.5495\, \text{cm}^0$ and $s = 23.5182\, \text{cm}$ we have:

$$ h ^ {\,2} + \frac{ 2.5495 }{ 4 } = \left( 23.5182\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} + \frac{ 6.5 }{ 4 } = \left( 23.5182\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} + 1.625 = \left( 23.5182\, \text{cm} \right)^{2} $$ $$ h ^ {\,2} = 553.1079\, \text{cm}^2 - 1.625 $$ $$ h ^ {\,2} = 551.4829\, \text{cm}^2 $$ $$ h = \sqrt{ 551.4829\, \text{cm}^2 } $$$$ h = 23.4837\, \text{cm} $$

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