Calculate volume, two slant heights, lateral surface area, total surface area, base area, and lateral edge length of a right rectangular pyramid.
A right rectangular pyramid is a three-dimensional solid with a rectangular base and four triangular faces that meet at a single apex positioned directly above the center of the base. Because the base is a rectangle rather than a square, the four lateral faces form two pairs of congruent triangles, each pair with a different slant height.
This shape appears in architecture (hip roofs), packaging design, earthwork calculations, and decorative structures. Engineers use it to estimate fill volumes, while architects model roof geometry with rectangular-pyramid formulas. In geology, many natural landforms approximate rectangular pyramids when sediment accumulates on a rectangular footprint.
This calculator requires three inputs — base length, base width, and perpendicular height — and returns seven key measurements: volume, two distinct slant heights (one for each pair of opposite triangular faces), lateral surface area, total surface area, base area, and lateral edge length. A unit selector and adjustable precision let you match your engineering or classroom needs.
Eight preset buttons let you explore real-world examples instantly — from a textbook problem to an earthwork pile. A bar chart visualizes how much of the total surface area is lateral versus base, and a reference table compares several rectangular pyramids side by side for quick lookup.
A right rectangular pyramid has more moving parts than a square pyramid because two different slant heights appear as soon as the base length and width stop matching. This calculator is useful when you need the full geometry quickly: roof framing layouts, hopper and chute concepts, packaging prototypes, and stockpile estimates all depend on volume plus accurate face dimensions. Instead of recalculating each triangular face separately, you can enter the three core dimensions once and get the base area, both slant heights, total surface area, and lateral edge length in a single pass.
Volume = (1/3) × l × w × h Slant Height (length face) = √(h² + (w/2)²) Slant Height (width face) = √(h² + (l/2)²) Lateral SA = l × slant_L + w × slant_W Total SA = Lateral SA + l × w Lateral Edge = √(h² + (l/2)² + (w/2)²)
Result: Volume = 40 cm³, Slant Heights ≈ 5.385 cm and 5.831 cm, Total SA ≈ 79.63 cm²
Using the textbook preset values, the base area is 6 × 4 = 24 cm² and the volume is (1/3) × 24 × 5 = 40 cm³. The slant height on the faces with base 6 is √(5² + 2²) = √29 ≈ 5.385 cm, while the slant height on the faces with base 4 is √(5² + 3²) = √34 ≈ 5.831 cm. Lateral surface area is 6 × 5.385 + 4 × 5.831 ≈ 55.63 cm², so total surface area is 55.63 + 24 ≈ 79.63 cm².
A right rectangular pyramid is easy to misread if you treat it like a square pyramid. The apex sits above the center of the rectangular base, so the distance to the midpoint of a long side is different from the distance to the midpoint of a short side. That creates two distinct slant heights, which then affect the triangular face areas. If you need fabric panels, roof sheathing, or sheet-metal cuts, using the wrong slant height will throw off both material estimates and layout dimensions.
This geometry appears in pyramid skylights, hopper transitions, decorative caps, monument tops, and stockpiles laid out on rectangular footprints. In estimating work, the volume helps with fill or material quantity, while total surface area helps with coatings, cladding, or wrapping. Because the base is rectangular, the shape is also common in designs that must align to room dimensions, pallet sizes, or framing grids instead of a perfectly square footprint.
For homework and technical sketches, start by separating the measurements into base dimensions and perpendicular height. Then decide whether you need internal capacity, which uses volume, or exterior material, which uses surface area and slant heights. This calculator is especially useful for checking whether a near-square base can be simplified or whether the difference between length and width is large enough that the two face pairs need to be treated separately in drawings and estimates.
Slant height is the distance from the apex to the midpoint of a base edge (perpendicular to that edge). Lateral edge is the distance from the apex to a base corner.
Because the base is a rectangle (l ≠ w), the apex is a different distance from the midpoint of the length edge vs. the width edge, producing two distinct slant heights.
Rearrange V = (1/3)lwh to h = 3V / (lw). Use this as a practical reminder before finalizing the result.
The volume formula V = (1/3)Bh works for any pyramid as long as h is the perpendicular height. However, the slant-height and surface-area formulas assume a right pyramid (apex over center).
Surface area scales with the square of the linear dimension, so doubling each side multiplies total SA by 4. Volume scales with the cube, so it multiplies by 8.
Yes — the net consists of one rectangle (l × w) and four triangles. Their dimensions are the base edges and corresponding slant heights, so you can lay them out accurately.