To calculate the surface area of a square pyramid, we need to consider both the base area and the areas of the four triangular sides.
Step 1: Calculate the Base Area
The base of our pyramid is a square. The area \( A \) of a square can be calculated using the formula:
\[ A = s^2 \]
where \( s \) is the length of a side of the square. In this case, the base length \( s \) is 12 inches. Plugging in the value:
\[ A = 12^2 = 144 \text{ square inches} \]
Step 2: Calculate the Slant Height
Next, we need to calculate the slant height \( l \) of the pyramid. The slant height is the height of the triangular face of the pyramid and can be found using the following relationship. We can visualize a right triangle formed by:
- The height of the pyramid (18 inches),
- Half the base length (which is \( \frac{12}{2} = 6 \) inches),
- The slant height \( l \), which is the hypotenuse.
Using the Pythagorean theorem:
\[ l = \sqrt{(\text{height})^2 + \left(\frac{\text{base length}}{2}\right)^2} \]
Plugging in the values:
\[ l = \sqrt{18^2 + 6^2} = \sqrt{324 + 36} = \sqrt{360} \]
Simplifying \( \sqrt{360} \):
\[ \sqrt{360} = \sqrt{36 \times 10} = 6\sqrt{10} \text{ inches} \]
Step 3: Calculate the Area of the Triangular Faces
The area \( A_t \) of a single triangular face is given by:
\[ A_t = \frac{1}{2} \times \text{base} \times \text{height} \]
For each triangular face of our pyramid, the base is 12 inches (the same length as the base of the square), and the height is the slant height we just calculated, \( l = 6\sqrt{10} \):
\[ A_t = \frac{1}{2} \times 12 \times 6\sqrt{10} = 36\sqrt{10} \text{ square inches} \]
Since there are 4 triangular faces, the total area of the triangular faces \( A_{triangles} \) is:
\[ A_{triangles} = 4 \times A_t = 4 \times 36\sqrt{10} = 144\sqrt{10} \text{ square inches} \]
Step 4: Calculate the Total Surface Area
Finally, we can calculate the total surface area \( A_{total} \) of the pyramid by adding the area of the base to the area of the triangular faces:
\[ A_{total} = A_{base} + A_{triangles} \]
Substituting the areas we calculated:
\[ A_{total} = 144 + 144\sqrt{10} \]
Thus, the surface area of the square pyramid with a base length of 12 inches and a height of 18 inches is:
\[ \boxed{144 + 144\sqrt{10}} \text{ square inches} \]