Surface Area of a Rectangular Prism Calculator
Enter a box's length, width, and height in any one unit (inches, feet, cm, m) to get total surface area, lateral area, and a face-by-face breakdown you can copy straight into homework.
Example: with Length (l) 10 · Width (w) 6 · Height (h) 4 · Unit generic units → Total surface area: 248 square units.
- Lateral area (4 sides only)128 square units
- Face-by-face breakdown2 × 60 + 2 × 40 + 2 × 24 = 248
- Base area (lw)60 square units (top and bottom each)
Computed by the calculator below using its default values. Change any input to see your own numbers.
SA = 2(lw + lh + wh): three different rectangles, each counted twice.
Why 2(lw + lh + wh) works
A box has six faces, but only three different ones: top/bottom (l × w), front/back (l × h), and the two ends (w × h). Compute each rectangle once, add them, and double the sum. The default 10 × 6 × 4 box gives 60 + 40 + 24 = 124, so the total surface area is 248 square units.
That pairing is also the fastest error check on homework: if your answer is odd, or you cannot split it into three doubled products, something slipped.
Total versus lateral
Total surface area covers all six faces — the number for wrapping paper, powder coating, or heat-loss estimates. Lateral area, 2h(l + w), covers just the four upright sides: the perimeter of the base times the height. Use lateral for painting walls without the ceiling and floor, or wrapping a label band around a carton.
How it’s calculated
SA = 2(lw + lh + wh) — three distinct rectangles, each appearing twice. Lateral area = 2h(l + w) covers only the four vertical faces; base area = lw. Values are rounded to 3 decimals and reported in the square of your chosen unit.
Treats the box as closed; an open-top container needs SA minus one l × w face.
Worked examples
| l × w × h | Face areas (lw, lh, wh) | Total surface area |
|---|---|---|
| 10 × 6 × 4 | 60, 40, 24 | 248 |
| 5 × 5 × 5 | 25, 25, 25 | 150 |
| 12 × 3 × 7 | 36, 84, 21 | 282 |
| 2 × 2 × 8 | 4, 16, 16 | 72 |
Computed with SA = 2(lw + lh + wh).
Common mistakes
- Forgetting the factor of 2 — each of the three distinct faces appears twice.
- Multiplying l × w × h — that is volume, not surface area; SA needs the pairwise products.
- Using total SA when the job only covers the sides; lateral area is 2h(l + w).
- Mixing units — one dimension in cm among inches corrupts all six faces.
Frequently asked questions
What is the surface area formula for a rectangular prism?
SA = 2(lw + lh + wh). Find the three distinct face areas — length × width, length × height, width × height — add them, and double, because each face has an identical twin on the opposite side.
What is lateral surface area?
The four side faces without the top and bottom: 2h(l + w), which is the base perimeter times the height. It is the right number for wall paint, labels, and siding.
Why pairwise products instead of multiplying all three edges?
Area is two-dimensional, so each face uses exactly two of the three edges. Multiplying all three gives cubic units — volume — which is the most common mix-up on this topic.
Does doubling one edge double the surface area?
No. Only the four faces that touch that edge grow; the two faces perpendicular to it stay the same. Doubling h in the default example moves SA from 248 to 376, not 496.