STEP 1: find side $ a $

To find side $ a $ use formula:

$$ AL = 3 \cdot a \cdot l $$

After substituting $AL = 500\, \text{cm}$ and $l = 24\, \text{cm}$ we have:

$$ 500\, \text{cm} = 3 \cdot a \cdot \left( 24\, \text{cm} \right)^{4} $$$$ 500\, \text{cm} = 72\, \text{cm} \cdot a $$$$ a = \dfrac{ 500\, \text{cm}}{ 72\, \text{cm} } $$$$ a \approx 2.2105 $$

STEP 2: find area $ A $

To find area $ A $ use formula:

$$ A = \frac{ a^2 \sqrt{3}}{2} + 3 a l $$

After substituting $a = 2.2105\, \text{cm}^0$ and $l = 24\, \text{cm}$ we have:

$$ A = \frac{ 2.2105^2 \sqrt{3}}{2} + 3 \cdot 2.2105 \cdot 24\, \text{cm} $$$$ A = \frac{ 4.8862 \sqrt{3}}{2} + 3 \cdot 53.0515\, \text{cm} $$$$ A = \frac{ 8.4632}{2} + 159.1546\, \text{cm} $$$$ A = 4.2316 + 159.1546\, \text{cm} $$$$ A = 163.3862 $$

Triangular Prism Calculator