What Is a Rectangular Pyramid?
A rectangular pyramid is a three-dimensional solid with a rectangular base and four triangular faces that meet at a single point called the apex. The apex sits directly above the center of the base in a right rectangular pyramid.
Formulas
Volume
One-third of the base area times the height.
Slant Height
Distance from the midpoint of a base edge to the apex.
Surface Area
Base area plus the area of all four triangular faces.
How the Calculation Works
Volume
The volume of any pyramid is one-third the area of its base times its perpendicular height. For a rectangular pyramid: V = (1/3) × l × w × h. This is exactly one-third the volume of a rectangular prism with the same base and height.
Slant Height
A rectangular pyramid has two different slant heights because the base is not square (unless l = w). The slant height along the length side runs from the midpoint of a width edge to the apex: s_l = √(h² + (w/2)²). Similarly for the width side: s_w = √(h² + (l/2)²).
Surface Area
The total surface area includes the rectangular base and the four triangular lateral faces. The lateral faces come in two pairs: two triangles with base l and slant height s_w, and two triangles with base w and slant height s_l.
Real-World Examples
- Architecture: Pyramid-shaped roofs and decorative structures.
- Packaging: Pyramid-shaped gift boxes and display stands.
- Engineering: Funnels and hoppers with rectangular openings.
- History: Ancient pyramids (though mostly square-based) follow the same volume principle.