### Video Transcript

Find the total surface area of the given regular pyramid and approximate the result to the nearest hundredth.

We’re asked to find the total surface area of this regular pyramid. A regular pyramid has a regular polygon on its base. In this case, the base has four sides, so it is a regular quadrilateral, by which we mean a square. To find the total surface area of this pyramid, we need to find the area of its square base and the area of each of its lateral faces. These are the triangular faces that connect each edge of the square base to the vertex of the pyramid. And as the pyramid is regular, they are congruent. Let’s find the area of the base first. As we said, it is a square, so its area is equal to its side length squared. It’s 32 squared, which is 1,024. And the units for this will be square centimeters.

Next, we need to consider the lateral area, the area of each of the triangular faces. We know that the area of a triangle is equal to its base multiplied by its perpendicular height over two. The base of these triangles is given in the diagram. It’s the side length of the square, which is 32 centimeters. But what about the perpendicular height? In the context of the lateral faces of a pyramid, this height has another specific name. It’s called the slant height of the pyramid. We need to be careful because the height we’ve been given in the diagram of 37 centimeters is not the slant height. It’s the perpendicular height of the pyramid.

We can, however, use this to calculate the slant height. There is a right triangle formed by the slant height, the perpendicular height, and this line here, which connects the midpoint of one edge of the base to the center of the base. This line is parallel to the sides of the square. And as it starts from the center, its length is half the side length of the square. It’s 32 over two, which is 16 centimeters. As we know two of the side lengths in a right triangle, we can calculate the third side length by applying the Pythagorean theorem. This states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. In our triangle, the side of length 𝐿 centimeters, where 𝐿 is representing the slant height of the pyramid, is the hypotenuse. So we have the equation 𝐿 squared equals 37 squared plus 16 squared. This simplifies to 𝐿 squared equals 1,369 plus 256, which is 1,625. 𝐿 is therefore equal to the square root of 1,625, which in simplified form is five root 65.

We’ve now found that the slant height of the pyramid, which is the perpendicular height of each of its lateral triangular faces, are five root 65 centimeters. Using the formula base times perpendicular height over two, the area of each of these triangles is 32 times five root 65 over two, and there are four of them. The lateral area of the pyramid simplifies to 320 root 65 square centimeters.

The total surface area is the sum of the base area and the lateral area, 1,024 plus 320 root 65, which as a decimal is equal to 3,603.9224 continuing. We’re asked to round the answer to the nearest hundredth. And as the number in the third decimal place is a two, we round down to 3,603.92. We’ve found that the total surface area of the given regular pyramid approximated to the nearest hundredth is 3,603.92 square centimeters.