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Geometry, coordinates, surveying and trigonometry
Total surface area of a right prism
Total surface of a closed right prism: two end faces of area A, plus the side faces whose combined width is the end perimeter P and whose length is h.
- Ref
- GEO-17
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- S [cm²]
A closed right prism has two end faces, each of area A, and side faces that together make a rectangle as wide as the end perimeter P and as long as h. So S = 2A + Ph. For an open end subtract that A. The lateral term assumes a right prism, with the sides at 90° to the ends.
Total surface of a closed right prism: two end faces of area A, plus the side faces whose combined width is the end perimeter P and whose length is h.
Calculator
- Area of the two ends, 2A24 cm²
- Area of the sides, Ph70 cm²
- Volume, Ah60 cm³
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A closed prism with ends of 12 cm² and perimeter 14 cm, 5 cm long, has 94 cm² of surface, of which 74.5 % is on the sides.
- The surface per unit volume is 1.567 per cm, which matters for heat loss and coating cost.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Length of the prism | 0.5 to 30 cm | 31 | 444 | 413 | +10% | +7.45 % |
| P Perimeter of the end | 2 to 80 cm | 34 | 424 | 390 | +10% | +7.45 % |
| A Area of one end | 1 to 100 cm² | 72 | 270 | 198 | +10% | +2.55 % |
Sweep of Length of the prism
| Length of the prism (cm) | Total surface area (cm²) |
|---|---|
| 0.5 | 31 |
| 4.188 | 82.63 |
| 7.875 | 134.3 |
| 11.56 | 185.9 |
| 15.25 | 237.5 |
| 18.94 | 289.1 |
| 22.63 | 340.8 |
| 26.31 | 392.4 |
| 30 | 444 |
Sweep one variable
Variables and how each one contributes
Total surface area
What the equation gives you. Everything else feeds it.
Area of one end
The area of one end face, in cm². Adds twice as much surface, because there are two ends. Raising it by 10 % (to 13.2) moves the result by +2.55 %, from 94 to 96.4 cm².
Perimeter of the end
The perimeter of an end face, in cm. Adds side area in proportion to the length. Raising it by 10 % (to 15.4) moves the result by +7.45 %, from 94 to 101 cm².
Length of the prism
The length of the prism, in cm. Adds side area in proportion to the perimeter. Raising it by 10 % (to 5.5) moves the result by +7.45 %, from 94 to 101 cm².
Worked example with the values above
- Ends: 2A = 2 × 12 = 24 cm².
- Sides: P h = 14 × 5 = 70 cm².
- S = 24 + 70 = 94 cm².
Assumptions and limits
- Solids with flat or smooth faces and a uniform cross-section where stated.
- All lengths in the same unit, with areas and volumes in the matching square and cubic units.
Common mistakes
- Using the slant height in place of the perpendicular height.
- Forgetting one end face in a closed prism, or counting an end that is open.
- Mixing cm² with cm³, or mm with cm, in one formula.
Statistics and uncertainty
Volumes scale with the cube of a length and surfaces with the square, so a 1 % error in a length gives about 3 % in a volume and 2 % in a surface.
Where it is used
- Material, paint and coating quantities for bars, tanks, hoppers and domes.
- Estimating volumes, masses and capacities from drawings.
Maths you need
- Volume: the space inside a solid, in cubic units
- Cross-section: the shape you see cutting the solid
- Surface area: the total area of all outer faces