Jim is building a model of a square pyramid for a class project. The side length of the square base is 12 inches and the slant height of the pyramid is 20 inches. What is the surface area of the model pyramid?
To find the surface area of the square pyramid, we need to calculate the area of the base and the area of the four triangular faces.
1. Area of the base:
The area of a square is calculated by multiplying the side length by itself.
Area of the base = 12 inches x 12 inches = 144 square inches
2. Area of the triangular faces:
Each triangular face of the pyramid is an isosceles triangle with a base length equal to the side length of the square base (12 inches) and a slant height of 20 inches. We can use the Pythagorean theorem to find the height of the triangle:
(height)^2 + (base/2)^2 = slant height^2
(height)^2 + (6 inches)^2 = 20 inches^2
(height)^2 + 36 = 400
(height)^2 = 400 - 36
(height)^2 = 364
height = sqrt(364)
height ≈ 19.1 inches
The area of each triangular face can be calculated using the formula for the area of a triangle:
Area of a triangle = 1/2 x base x height
Area of each triangular face = 1/2 x 12 inches x 19.1 inches = 114.6 square inches
3. Now, we can calculate the total surface area of the square pyramid by adding the area of the base and the four triangular faces:
Total surface area = Area of base + 4 x Area of each triangular face
Total surface area = 144 square inches + 4 x 114.6 square inches
Total surface area = 144 square inches + 458.4 square inches
Total surface area = 602.4 square inches
Therefore, the surface area of Jim's model pyramid is 602.4 square inches.