find the surface area of the regular pryamid
A pyramid with a square base is shown. The sides of the square base measure 4 feet. A dashed line which is labeled 7 feet extends from the middle of a side of the base along a triangular surface to the vertex at the top of the pyramid.
To find the surface area of the pyramid, we need to find the area of the base (the square) and the total area of the 4 triangular faces.
First, let's find the area of the base (the square):
The length of one side of the square is 4 feet, so the area of the base is 4 * 4 = <<4*4=16>>16 square feet.
Next, we need to find the area of one of the triangular faces, and then multiply it by 4 to get the total area of all the triangular faces.
The height of each triangular face can be found using the Pythagorean theorem. In this case, since we have a right triangle with hypotenuse (the dashed line) of 7 feet, and one leg (half of the square's side) of 2 feet, the height can be found by solving for the other leg in the equation:
h^2 + 2^2 = 7^2
h^2 + 4 = 49
h^2 = 45
h = sqrt(45) = 3 * sqrt(5)
Now we can find the area of one triangle:
The base of the triangle is 4 feet and the height is 3 * sqrt(5), so the area is:
Triangle_area = 0.5 * base * height
Triangle_area = 0.5 * 4 * (3 * sqrt(5))
Triangle_area = 6 * sqrt(5)
Since there are 4 triangles, the total area of the triangles is 4 * (6 * sqrt(5)) = 24 * sqrt(5) square feet.
Now, let's add the area of the base to the area of the triangles to find the surface area of the pyramid:
Surface_area = Base_area + Triangles_area
Surface_area = 16 + 24 * sqrt(5) ≈ 53.67 square feet.
So the surface area of the regular pyramid is approximately 53.67 square feet.