Rain made an error.
The expression should be [tex]3\left(\frac{1}{2}\times10\times \bf{12}\right)+ \left(\frac{1}{2}\times10 \times 8.7\right)[/tex]
Note the term in bold which is 12. I've replaced 13 with this value. This is because the height of each lateral side is 12 inches, instead of 13 inches.
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The steps needed to be taken are therefore:
[tex]3\left(\frac{1}{2}\times10\times 12\right)+ \left(\frac{1}{2}\times10 \times 8.7\right)\\\\\\3\left(5\times 12\right)+ \left(5 \times 8.7\right)\\\\\\3\left(60\right)+ 43.5\\\\\\180+ 43.5\\\\\\223.5\\\\\\[/tex]
This means the triangular pyramid's surface area should be 223.5 square inches.