Find the lateral area and surface area of the square pyramid
13in
10in
To find the lateral area of a square pyramid, we first need to find the slant height of the pyramid. We can use the Pythagorean theorem to find the slant height:
h^2 = 10^2 + (1/2 * 13)^2
h^2 = 100 + 84.5
h^2 = 184.5
h ≈ √184.5
h ≈ 13.58 inches
Now that we have the slant height, we can find the lateral area of the pyramid. The lateral area is given by the formula:
Lateral Area = 0.5 * Perimeter of base * Slant height
The perimeter of the base of the square pyramid is 4 * 13 = 52 inches. Therefore, the lateral area is:
Lateral Area = 0.5 * 52 * 13.58
Lateral Area ≈ 353.56 square inches
To find the total surface area of the pyramid, we also need to consider the area of the base. The base of the pyramid is a square with side length 13 inches, so the base area is:
Base Area = 13^2 = 169 square inches
To get the total surface area, we sum the lateral area and the base area:
Surface Area = Lateral Area + Base Area
Surface Area = 353.56 + 169
Surface Area ≈ 522.56 square inches
Therefore, the lateral area of the square pyramid is approximately 353.56 square inches, and the total surface area is approximately 522.56 square inches.