Rectangular Prism Surface Area Calculator (Box)

SA = 2 × (l × w + l × h + w × h)

Six rectangular faces, three matched pairs. Top and bottom each = l × w; front and back each = l × h; left and right each = w × h. Sum the three pair areas and multiply by 2.

Worked example: painting a room, interior surfaces only A bedroom 12 ft × 14 ft × 9 ft (length × width × height), painting walls and ceiling but not the floor.

Walls only: 2 × (12 × 9) + 2 × (14 × 9) = 216 + 252 = 468 sq ft. Ceiling: 12 × 14 = 168 sq ft. Total: 636 sq ft to cover.

Subtract two windows at 15 sq ft each and a door at 20: 636 − 50 = 586 sq ft of net paint area. Two coats is 1,172 sq ft of coverage. At the 350 sq ft per gallon end of the range that is 3.35 gallons, so buy 4 and keep the rest for touch-ups. Three gallons only works if your paint genuinely covers 400 sq ft a coat, and few do on a first coat over a colour change.

Where rectangular surface area matters:

• Interior wall paint. Sum the wall and ceiling rectangles, subtract openings, then multiply by number of coats.
• Wrapping paper for boxes. Multiply geometric SA by 1.2-1.5 for overlap.
• Insulation for shipping containers. Refrigerated boxes need foam coverage on all six faces.
• Sheet metal HVAC ductwork. Rectangular duct is sold by the linear foot, so divide the surface area by the unrolled width to get the run you need.
• Heat exchange surface (radiators, cold plates). Heat transfer scales with contact area.
• Aquarium glass. A 20 gal long tank measures 30" × 12" × 12" and needs five panes, not six, because there is no top: bottom 30 × 12 = 360, front and back 360 each, and two ends at 12 × 12 = 144 each. That comes to 1,368 sq in of glass.
• Solar panel framing. Aluminium extrusion runs the perimeter; the glass is one of the six rectangles.

Cost estimation shortcuts:

Cube as a special case:

If l = w = h = s, the formula gives 2 × (s² + s² + s²) = 6s². Matches the cube formula. ✓

The trick of measuring just one edge:

For a “near-cube” box where l ≈ w ≈ h, you can estimate surface as approximately 6 × edge². For a 30" × 32" × 34" box: actual SA = 2 × (960 + 1020 + 1088) = 6,136 sq in. Cube approximation with average edge 32: 6 × 1024 = 6,144 sq in. Within 0.13%. Useful for back-of-envelope estimates.

Sanity check:

• l = w = h: collapses to 6s² (cube). ✓
• h = 0: the box flattens to a sheet and SA = 2lw. The four faces that used h vanish, and what survives is the top and bottom of a flat rectangle counted on both sides. ✓
• Of every box with a given volume, the cube has the least surface. A 1,000 cm³ cube is 600 cm² of skin; the same volume as a 100 × 10 × 1 slab is 2,220 cm². That is why packaging drifts toward cubes and why a long thin room costs more to heat.