Rectangular Pyramid Surface Area Calculator

A rectangular pyramid has a rectangular base and four triangular faces — two congruent pairs (NOT all four congruent like a square pyramid).

SA = l × w + l × √(h² + (w/2)²) + w × √(h² + (l/2)²)

Where:

• l, w = base length and width
• h = perpendicular height from base center to apex
• The two slant heights are different — one for the long-side triangles, one for the short-side triangles.

Worked example — hipped roof on a 30 × 50 ft house (true pyramid hip, unusual): A 30 × 50 ft single-family house with a true pyramidal hipped roof (apex over center, no ridge — unusual but possible). Rise to apex h = 12 ft. Slant for long-side faces: l_long = √(144 + 625) = √769 ≈ 27.73 ft. Slant for short-side faces: l_short = √(144 + 225) = √369 ≈ 19.21 ft. Long-side triangular faces (2 of them): 2 × (½ × 50 × 27.73) = 1,386.5 sq ft. Short-side triangular faces (2 of them): 2 × (½ × 30 × 19.21) = 576.3 sq ft. Total roof area (no base): 1,962.8 sq ft.

That’s 19.6 “squares” of shingles (each square = 100 sq ft). Buy 21-22 squares to allow for waste and ridge caps.

Note: most real “hipped” roofs have a ridge, not a single apex. A 50 ft long house typically has a horizontal ridge running most of the length, with hip ends only at the gables. The true-pyramid case requires a square or near-square footprint.

Where rectangular pyramid surface matters:

• Hipped roofs on rectangular small buildings. Garden sheds, garages, small additions sometimes have pure pyramidal hips.
• Pyramidal pavilion roofs. Park gazebos on rectangular bases.
• Pyramid-shaped greenhouses. Conservatory tops on rectangular bases.
• Hopper bottoms in industrial silos with rectangular cross-sections. Sheet metal for the hopper walls.
• Architectural display pedestals. Pyramid-shaped concrete or marble bases for sculptures.

Two different slant heights, two different formulas:

This is what trips people up. A rectangular pyramid (non-square base) has FOUR triangular faces, but they aren’t all the same shape:

• Two triangular faces have base = l (the long base edge) and slant l_long = √(h² + (w/2)²).
• Two triangular faces have base = w (the short base edge) and slant l_short = √(h² + (l/2)²).

Total lateral surface = l × l_long + w × l_short.

Get this right by carefully tracking which edge each face sits against.

Open vs. closed:

• Closed (with base): SA = l × w + lateral. Used for solid pyramid models, architectural sculptures.
• Open (no base): lateral only. Used for tents, hipped roofs, pavilion canopies.

Sanity check:

• l = w: collapses to square pyramid (both slant heights become equal). ✓
• h = 0: pyramid flattens; lateral = 0; SA = base area. ✓
• A 10 × 10 × 12 case: l_long = l_short = √(144 + 25) = √169 = 13. Lateral = 10×13 + 10×13 = 260. Base = 100. Total = 360. Matches square pyramid 10² + 2×10×13 = 100 + 260 = 360. ✓