To calculate the surface area of a square pyramid, we need to use the formula for the surface area, which is:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
Step 1: Calculate the Base Area
The base of the pyramid is a square, so the area of the base can be calculated using the formula for the area of a square:
\[ \text{Base Area} = \text{side}^2 \]
Given that the length of the side of the base is 12 inches, we plug this value into the formula:
\[ \text{Base Area} = 12^2 = 144 \text{ square inches} \]
Step 2: Calculate the Slant Height
To calculate the lateral area, we need the slant height of the pyramid. The slant height can be found using the Pythagorean theorem. In a square pyramid, the slant height (l) forms a right triangle with half the base length (a) and the height (h) of the pyramid. The relationship is:
\[ l = \sqrt{h^2 + a^2} \]
Here, \( h = 18 \) inches (height of the pyramid) and \( a = \frac{12}{2} = 6 \) inches (half of the base). Plugging these values in, we get:
\[ l = \sqrt{18^2 + 6^2} \] \[ l = \sqrt{324 + 36} \] \[ l = \sqrt{360} \] \[ l = \sqrt{36 \times 10} = 6\sqrt{10} \text{ inches} \]
Step 3: Calculate the Lateral Area
The lateral area of a pyramid can be calculated using the formula:
\[ \text{Lateral Area} = \frac{1}{2} \times \text{Perimeter of Base} \times \text{Slant Height} \]
The perimeter \( P \) of the base (which is a square) is:
\[ P = 4 \times \text{side} = 4 \times 12 = 48 \text{ inches} \]
Now we can find the lateral area:
\[ \text{Lateral Area} = \frac{1}{2} \times P \times l = \frac{1}{2} \times 48 \times 6\sqrt{10} \] \[ = 24 \times 6\sqrt{10} \] \[ = 144\sqrt{10} \text{ square inches} \]
Step 4: Calculate the Total Surface Area
Now we can find the total surface area by summing the base area and the lateral area:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \] \[ = 144 + 144\sqrt{10} \text{ square inches} \]
Final Result
Thus, the total surface area of the pyramid is:
\[ \text{Surface Area} = 144 + 144\sqrt{10} \text{ square inches} \]
This is the complete solution to finding the surface area of the square pyramid with a base length of 12 inches and a height of 18 inches.