Question

What is the correct formula for calculating the surface area of a rectangular pyramid with given base sides and slant heights?

Answer 1

To calculate the surface area of a rectangular pyramid, you need to consider both the area of the base and the areas of the triangular faces.

Let's denote:
- \( a \) = length of the base
- \( b \) = width of the base
- \( l_1 \) = slant height of the triangular face corresponding to the side with length \( a \)
- \( l_2 \) = slant height of the triangular face corresponding to the side with width \( b \)

The surface area \( SA \) of the rectangular pyramid can be calculated using the following formula:

\[
SA = A_{\text{base}} + A_{\text{lateral}}
\]

1. **Calculate the area of the base**:

The area of the rectangular base is given by:
\[
A_{\text{base}} = a \times b
\]

2. **Calculate the area of the triangular faces**:

The pyramid has four triangular faces:

- Two triangles with a base of \( a \) and height \( l_1 \):
\[
A_{\text{triangles with base } a} = 2 \times \left(\frac{1}{2} \times a \times l_1\right) = a \times l_1
\]

- Two triangles with a base of \( b \) and height \( l_2 \):
\[
A_{\text{triangles with base } b} = 2 \times \left(\frac{1}{2} \times b \times l_2\right) = b \times l_2
\]

Therefore, the total lateral surface area is:
\[
A_{\text{lateral}} = (a \times l_1) + (b \times l_2)
\]

3. **Combine the areas**:

Putting it all together, you get the surface area of the rectangular pyramid:
\[
SA = a \times b + (a \times l_1) + (b \times l_2)
\]

This formula gives you the total surface area encompassing the base area and the areas of the triangular sides.