Question

what is the surface area of this rectangular pyramid? 16 mm 16 mm 13 mm square millimeters submit

Explanation:

Step1: Identify the base and lateral faces

The base is a square with side length \( 16 \) mm. The lateral faces are four congruent triangles? Wait, no, wait. Wait, the base is a square (since length and width are both 16 mm), and the triangular faces: wait, the slant height? Wait, the given 13 mm—wait, maybe the base is a square with side 16 mm, and the triangular faces have a slant height? Wait, no, wait. Wait, the rectangular pyramid—wait, if the base is a square (16x16), then the lateral faces are four triangles? Wait, no, a rectangular pyramid has a rectangular base, but here both length and width are 16, so it's a square pyramid (a type of rectangular pyramid where base is square). So the surface area is the area of the base plus the area of the four lateral triangular faces.

Wait, the base area: \( 16 \times 16 = 256 \) square mm.

Now, the lateral faces: each triangular face has a base of 16 mm and a height (slant height) of 13 mm? Wait, the diagram shows 13 mm as the slant height? Wait, the triangular face: the area of one triangular face is \( \frac{1}{2} \times base \times slant\ height \). Since there are four such faces? Wait, no, wait: in a square pyramid, there are four triangular faces? Wait, no, a square pyramid has a square base and four triangular lateral faces. Wait, but if the base is square (16x16), and each triangular face has base 16 and height 13, then the area of one triangular face is \( \frac{1}{2} \times 16 \times 13 = 104 \) square mm. Then four of them: \( 4 \times 104 = 416 \) square mm. Then total surface area is base area plus lateral surface area: \( 256 + 416 = 672 \) square mm. Wait, is that correct?

Wait, let's recheck. The base is a square with side 16, so area \( 16 \times 16 = 256 \). The lateral faces: each is a triangle with base 16 and height 13. There are four lateral faces? Wait, no, wait: in a square pyramid, there are four triangular faces, each with base equal to the side of the square and height (slant height) as given. So each triangle area: \( \frac{1}{2} \times 16 \times 13 = 104 \). Four triangles: \( 4 \times 104 = 416 \). Then total surface area: \( 256 + 416 = 672 \).

Step2: Calculate base area

Base is a square with side \( 16 \) mm.
Base area \( = 16 \times 16 = 256 \) \( \text{mm}^2 \).

Step3: Calculate lateral surface area

Each lateral face is a triangle with base \( 16 \) mm and height \( 13 \) mm.
Area of one triangular face \( = \frac{1}{2} \times 16 \times 13 = 104 \) \( \text{mm}^2 \).
There are 4 lateral faces, so lateral surface area \( = 4 \times 104 = 416 \) \( \text{mm}^2 \).

Step4: Calculate total surface area

Total surface area \( = \) base area \( + \) lateral surface area \( = 256 + 416 = 672 \) \( \text{mm}^2 \).

Answer:

\( 672 \)