Right Rectangular Pyramid Calculator
Calculate volume, surface areas, slant heights, diagonal, and edge length from the base dimensions and perpendicular height.
Pyramid dimensions
Required positive length; dot decimals are accepted.
Required positive width in the selected length unit.
Perpendicular distance from the base center to the apex.
Changing this setting converts all three current dimensions.
Live results
Volume (V)
192 in³
Total surface area (A)
222.849 in²
Base surface area (Aᵦ)
48 in²
Lateral surface area (Aₗ)
174.849 in²
Lateral edge (d)
13 in
Surface area composition
| Surface component | Area | Share of total |
|---|
The two parts are mutually exclusive: base area plus lateral area equals total surface area.
Geometry details
| Measurement | Meaning | Formula | Value |
|---|
Slant height over side a is the altitude of a triangular face whose base is a; the corresponding height depends on half of b. The roles reverse for the face over side b.
How to use the right rectangular pyramid calculator
What this calculator does
This calculator solves a right rectangular pyramid from three perpendicular measurements: Base length (a), Base width (b), and Pyramid height (H). A right pyramid has its apex directly above the center of the rectangular base. From those inputs, the calculator finds the exact geometric identities for volume, base area, lateral area, total surface area, base diagonal, lateral edge, two slant heights, and the areas of the two distinct triangular face pairs. It does not estimate material waste, wall thickness, construction tolerances, or the geometry of an oblique pyramid.
When to use it
Use it to check homework or design calculations, estimate the volume of a pyramid-shaped solid, determine how much exterior material covers the base and four side faces, or compare the proportions of rectangular and square pyramids. It is also useful when a drawing gives only the base dimensions and perpendicular height but asks for hidden lengths such as a slant height or lateral edge.
How to calculate
- The calculator opens with a complete demonstration: 6 in by 8 in with a 12 in height. Its results and a validated example XLSX workbook are available immediately.
- Replace Base length (a), Base width (b), and Pyramid height (H) with positive values. Use a dot for decimals. Commas are accepted only as conventional thousands separators, so 1,500.5 is valid while ambiguous 1,5 is rejected.
- Choose the Length unit. The calculator converts all three current dimensions, then reports lengths, square units for areas, and cubic units for volume.
- Read Volume (V) as the primary result. Review the surface-area cards, composition chart, and geometry table for the remaining identities.
- Select Download Excel to export the current validated inputs and results as a real XLSX workbook. Reset clears the demonstration and calculated state; export remains unavailable until all required dimensions are entered again.
Input guide
Base length (a) is a required positive decimal in the selected unit; 6 in is the example. Increasing it raises base area, volume, diagonal, edge length, one face pair, and total area. Do not enter an area or a negative length. Base width (b) is the other required base side; 8 in is the example. It has the same format and restrictions, but it drives the opposite slant-height relationship. Pyramid height (H) is the required perpendicular center-to-apex distance; 12 in is the example. A larger height increases volume, lateral edge, both slant heights, and lateral area. Do not substitute a slant height for H. Length unit accepts centimeters, meters, inches, or feet and converts the current entries rather than merely relabeling them.
Output guide
Volume (V) is cubic capacity and equals one third of base area times perpendicular height; it is an exact identity for this model. Total surface area (A) is the base plus all four triangular side faces. Base surface area (Aᵦ) is a × b. Lateral surface area (Aₗ) excludes the base and combines the two equal face pairs. Lateral edge (d) runs from the apex to any base corner. The geometry table also reports Base diagonal, Slant height over side a, Slant height over side b, Face area over side a, and Face area over side b. Zero is not a valid finished pyramid dimension, and all valid outputs are positive. The surface composition chart compares base and lateral area in the same square unit; its two shares always total 100%.
Worked example
For a = 6 in, b = 8 in, and H = 12 in, the base area is 6 × 8 = 48 in². The volume is (48 × 12) ÷ 3 = 192 in³. The base diagonal is √(6² + 8²) = 10 in, and the lateral edge is √(12² + (10 ÷ 2)²) = 13 in. The face slant heights are √(12² + (8 ÷ 2)²) ≈ 12.649 in over side a and √(12² + (6 ÷ 2)²) ≈ 12.369 in over side b. Those face dimensions produce a lateral area of about 174.849 in² and a total surface area of 222.849 in², matching the first-open results and workbook.
Learn more
OpenStax derives the general pyramid volume formula V = Ah/3. For terminology, MathWorld explains the defining relationship between a polygonal base, triangular faces, apex, and a right pyramid.
Formulas and interpretation
The calculation starts with the rectangular base, then applies the Pythagorean theorem to right-triangle cross-sections. Each side-face pair has its own slant height unless a = b, in which case the pyramid has a square base and the two values coincide.
For the two lateral face types, sₐ = √(H² + (b/2)²) and sᵦ = √(H² + (a/2)²). Therefore Aₗ = a·sₐ + b·sᵦ. The corner edge is d = √(H² + (a² + b²)/4). These relationships assume a perfectly centered apex and planar faces.
Keep every input in one length unit. Area scales with the square of a uniform size change, while volume scales with the cube. Doubling a, b, and H therefore multiplies total area by four and volume by eight. NIST's dimensional-analysis guidance explains why length, area, and volume require first-, second-, and third-power conversion factors.
Displayed values are rounded to a practical precision, while calculations and the Excel workbook retain canonical numeric values.