Right Rectangular Pyramid Calculator

A right rectangular pyramid has a rectangular base and an apex directly above the base's center. Its perpendicular height h drops from the apex to that center. The base length l and width w may differ, producing two pairs of congruent triangular faces. Because the apex is centered, the pyramid is right even when its base is not square. Base area is simply l × w. Volume follows the general pyramid rule: one third of base area times perpendicular height, so V = lwh / 3. A pyramid therefore occupies one third of a rectangular prism with the same base and height. This formula uses the vertical height, not an edge or face slant height. Lateral surface area requires special care because a rectangular base generally creates two different face slant heights. The triangular faces whose bases have length l rise across half the width, giving slant height √(h² + (w/2)²). Their combined area is l times that slant height. The faces whose bases have width w rise across half the length, giving √(h² + (l/2)²), and their combined area is w times that value. Adding both pairs yields the complete lateral area. Total surface area adds the rectangular base. The calculator reports both slant heights so the geometry can be checked or used for fabrication. If length and width are equal, the base is square and both slant heights match. Do not confuse a face slant height, which runs from the midpoint of a base edge to the apex, with a lateral edge, which runs from a base corner to the apex. They form different right triangles and have different lengths. Use the same unit for all three inputs. Linear results retain that unit, areas use square units, and volume uses cubic units. The displayed results are rounded to six decimal places after calculations are performed at full floating-point precision. For practical construction, packaging, roofing, or material estimates, add allowances for thickness, seams, overlap, and waste after calculating the ideal geometry. Rectangular pyramids appear in architectural roofs, monuments, hoppers, packaging, and geometry exercises. Volume can estimate internal capacity or fill material. Lateral area estimates covering for the four triangular faces, while total area includes the bottom rectangle. A useful scaling check is that multiplying every dimension by a factor k multiplies slant heights by k, areas by k², and volume by k³. These relationships make unit and transcription errors easier to detect.