Calculate the surface area of the square pyramid that has a base length of 12 inches and a height of 18 inches. Show your work by plugging in the dimensions of the pyramid into the correct surface area formula. Explain each step of your solving process.

Answer 1

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.