Jim is building a model of a square pyramid for a class project. The side length of the square base is 20 inches and the slant height of the pyramid is 13 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 square base and the area of the four triangular faces.
1. Area of the square base:
The formula to find the area of a square is side length squared. Therefore, the area of the square base is 20 inches x 20 inches = 400 square inches.
2. Area of the four triangular faces:
The formula to find the area of a triangle is 1/2 x base x height. Since the base of the triangle is the side length of the square base, the height can be found using the Pythagorean theorem:
height^2 = slant height^2 - (side length/2)^2
height^2 = 13^2 - (20/2)^2
height^2 = 169 - 100
height^2 = 69
height ≈ 8.31 inches
Now, we can calculate the area of each triangular face:
Area = 1/2 x base x height
Area = 1/2 x 20 inches x 8.31 inches
Area ≈ 83.1 square inches
Since there are four triangular faces, the total area of the triangular faces is 4 x 83.1 = 332.4 square inches.
3. Total Surface Area:
The total surface area of the square pyramid is the sum of the area of the square base and the area of the four triangular faces:
Total Surface Area = 400 + 332.4 = 732.4 square inches
Therefore, the surface area of the model square pyramid is 732.4 square inches.