Enter the length, width and height of the box. The surface area ((length × width + length × height + width × height) × 2) is shown, along with conversions between m² and cm².
Use the same unit for all three and enter only numbers greater than 0 (numbers only, no units. For 20 cm, enter "20").
Result and figure
Enter the length, width and height in the fields on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the three lengths (length, width and height), and the surface area of the rectangular prism (the total area of its 6 faces) is calculated on the spot
- The result also shows the area in another unit: square feet (ft²) if you enter inches, and square inches (in²) if you enter feet. Switch "Units" to Metric to use cm² and m² instead
- The result is also drawn as a 3D shape. Drag it with the mouse to turn it around and see which faces were added up
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter the length, width and height in the same unit (all in inches, or all in feet). The surface areas of other solids, such as cubes, cylinders and spheres, have their own pages. For how much the box holds (its volume), see the rectangular prism volume page.
What is this calculation used for?
Estimating the wrapping paper for a gift (everyday life)
To wrap a box that is 12 in long, 9 in wide and 4 in tall, the surface area of the box is \((12 \times 9 + 12 \times 4 + 9 \times 4) \times 2 = 384\,\mathrm{in^2}\).
Real wrapping needs extra for folds and overlaps, so plan on about 1.3 to 1.5 times the surface area, roughly 500 to 580 in². This calculation tells you whether the paper you have is enough before you start.
Calculating the cardboard for a box (packaging and shipping)
A common medium moving box is 18 in by 18 in and 16 in tall. The cardboard it uses is its surface area, \((18 \times 18 + 18 \times 16 + 18 \times 16) \times 2 = 1800\,\mathrm{in^2}\), which is \(1800 \div 144 = 12.5\) ft² (the real blank also has flaps and a glue strip).
Makers who produce thousands of boxes look for sizes with the smallest surface area for the same volume, because a small difference in area adds up to a big difference in material cost.
Estimating the paint for the walls and ceiling of a room (DIY and remodeling)
Think of a room 12 ft long and 10 ft wide with an 8 ft ceiling as a box. To paint the 4 walls and the ceiling, the area is walls \((12 + 10) \times 2 \times 8 = 352\,\mathrm{ft^2}\) plus ceiling \(12 \times 10 = 120\,\mathrm{ft^2}\), for 472 ft² in all (the surface area without the floor; in practice you also subtract windows and doors).
A gallon of paint usually covers about 350 to 400 ft² per coat, and the can says how much, so once you know the area you can work out how many gallons to buy.
Thinking about heat moving in and out of a cooler or a building (keeping cool and insulation)
Heat moves in and out through the surface, so the amount of heat lost is roughly proportional to the surface area. A cooler that is 26 in long, 14 in wide and 15 in tall has \((26 \times 14 + 26 \times 15 + 14 \times 15) \times 2 = 1928\,\mathrm{in^2}\), about 13.4 ft², of surface touching the outside air.
For the same volume, a shape with less surface area (closer to a cube) keeps things cold longer, and the amount of insulation for appliances and buildings also depends on surface area. This calculation is the basis for those decisions.
Estimating the cost of plating or coating metal parts (manufacturing)
Plating and coating are often priced by the surface area treated, so a quote starts with a surface area calculation. A flat part that is 4 in long, 2 in wide and 1 in thick has a surface area of \((4 \times 2 + 4 \times 1 + 2 \times 1) \times 2 = 28\,\mathrm{in^2}\).
Plating 1,000 of these parts means treating 28,000 in², about 194 ft², in total. The same number feeds straight into production plans and the amount of chemicals needed.
Formulas and figures
Surface area of a rectangular prism
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\((\)
\(l\)
\(\times\)
\(w\)
\(+\)
\(l\)
\(\times\)
\(h\)
\(+\)
\(w\)
\(\times\)
\(h\)
\()\)
\(\times\)
\(2\)
In words
(symbols replaced with words)
⑤
\(S\): surface area
\(=\)
\((\)
①
\(l\): length
\(\times\)
②
\(w\): width
\(+\)
\(l\): length
\(\times\)
③
\(h\): height
\(+\)
\(w\): width
\(\times\)
\(h\): height
\()\)
\(\times\)
④
\(2\): faces in each matching pair
The formula in words
①
Start with the
\(l\): length
②
the
\(w\): width
③
and the
\(h\): height
. Multiply them two at a time to get the areas of the 3 kinds of faces, and add those areas
④
then multiply the sum by the
\(2\): faces in each matching pair
⑤
and you get the
\(S\): surface area
Quick example
The surface area of a box that is 3 in long, 4 in wide and 5 in tall is
\(S\): surface area
\(=\)
\((\)
length (3 in)
\(\times\)
width (4 in)
\(+\)
length (3 in)
\(\times\)
height (5 in)
\(+\)
width (4 in)
\(\times\)
height (5 in)
\()\)
\(\times\)
faces in each pair (2)
\((3 \times 4 + 3 \times 5 + 4 \times 5) \times 2 = (12 + 15 + 20) \times 2\)
\(= 47 \times 2 = 94\,\mathrm{in^2}\)
Key idea
Why does this formula give the surface area? A rectangular prism has 6 rectangular faces, but look closely and they come in matching pairs on opposite sides: the bottom and top (length × width), the left and right (length × height), and the front and back (width × height). So adding the 3 kinds of face areas and doubling the sum gives the total of all 6 faces.
Unlike volume (length × width × height), surface area adds up flat faces, so its unit is squared (in², ft²). Use surface area for how much paper covers a box, and volume for how much the box holds.
Base area × 2 + lateral area (surface area of a prism)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(S_1\)
\(\times\)
\(2\)
\(+\)
\(L\)
\(\times\)
\(h\)
In words
(symbols replaced with words)
⑤
\(S\): surface area
\(=\)
①
\(S_1\): area of the base
\(\times\)
②
\(2\): number of bases
\(+\)
③
\(L\): perimeter of the base
\(\times\)
④
\(h\): height
The formula in words
①
Take the
\(S_1\): area of the base
②
multiply it by the
\(2\): number of bases
to cover the bottom and the top
③
then add the
\(L\): perimeter of the base
④
times the
\(h\): height
(the lateral area)
⑤
and you get the
\(S\): surface area
Quick example
The surface area of a box that is 3 in long and 4 in wide (base area 12 in², base perimeter 14 in) and 5 in tall is
\(S\): surface area
\(=\)
area of the base (12 in²)
\(\times\)
number of bases (2)
\(+\)
perimeter of the base (14 in)
\(\times\)
height (5 in)
\(3 \times 4 = 12\,\mathrm{in^2} \qquad (3 + 4) \times 2 = 14\,\mathrm{in}\)
\(12 \times 2 + 14 \times 5 = 24 + 70 = 94\,\mathrm{in^2}\)
Key idea
On the net of the box, the 4 side faces join into one big rectangle whose width is the perimeter of the base and whose height is the height of the box. So the lateral area (the total area of the sides) is simply "perimeter of the base × height". Add the bottom and the top (area of the base × 2) and you have the surface area.
The nice thing about this view is that it works even when the base is not a rectangle. Whether the base is a triangle or a pentagon, the surface area of any prism is "area of the base × 2 + perimeter of the base × height". Many US textbooks write it as \(S = 2B + Ph\). The answer is, of course, the same as with the first formula.
Converting area units (in² to ft²)
Figure
Standard notation
(the usual math form)
\(S_{\mathrm{ft^2}}\)
\(=\)
\(S_{\mathrm{in^2}}\)
\(\div\)
\(144\)
In words
(symbols replaced with words)
③
\(S_{\mathrm{ft^2}}\): surface area in ft²
\(=\)
①
\(S_{\mathrm{in^2}}\): surface area in in²
\(\div\)
②
\(144\): square inches in 1 ft²
The formula in words
①
Take the
\(S_{\mathrm{in^2}}\): surface area in in²
②
divide it by
\(144\): square inches in 1 ft²
③
and you get the
\(S_{\mathrm{ft^2}}\): surface area in ft²
Quick example
A box that is 20 in by 30 in and 10 in tall has a surface area of 2200 in². In square feet, that is
surface area in ft²
\(=\)
surface area in in² (2200)
\(\div\)
square inches in 1 ft² (144)
\(2200 \div 144 \approx 15.28\,\mathrm{ft^2}\)
Key idea
\(1\,\mathrm{ft^2}\) is the area of a square that is 1 foot (12 inches) on each side, so it equals \(12 \times 12 = 144\,\mathrm{in^2}\). That is why you divide square inches by 144 to get square feet. The key is that the length factor 12 is squared; dividing by 12 is a common mistake.
Compare this with volume (\(1\,\mathrm{ft^3} = 12^3 = 1728\,\mathrm{in^3}\)) and you see one rule: areas use the square of the length factor, and volumes use the cube.
The surface area of a rectangular prism is (length × width + length × height + width × height) × 2, which is twice the sum of the areas of the 3 kinds of faces. It can also be read as "area of the base × 2 + perimeter of the base × height", a view that works for every prism. To convert square inches to square feet, just divide by 144.
Symbols and terms
Symbols
| \(S\) | ess | A common symbol for area. On this page it stands for the surface area of the rectangular prism. Think of "surface" to remember it (some books write SA). |
| \(l\) | ell | The length of the box, from the first letter of "length". |
| \(w\) | double-u | The width of the box, from the first letter of "width". |
| \(h\) | aitch | The height of the box, from the first letter of "height". |
| \(S_1\) | S sub one | The area of the base (the rectangle at the bottom). On this page, \(S_1 = l \times w\). The small 1 keeps it apart from the surface area \(S\). Many US textbooks write \(B\) for this. |
| \(L\) | capital L | The perimeter of the base (the distance around it). For a base that is a rectangle with length \(l\) and width \(w\), \(L = (l + w) \times 2\). Many US textbooks write \(P\) for this. |
| \(\mathrm{in^2}\) | square inches | A unit of area. A square that is 1 inch on each side has an area of 1 in². Do not mix it up with in³ (cubic inches), which is a unit of volume. |
| \(\mathrm{ft^2}\) | square feet | A unit of area. A square that is 1 foot on each side has an area of 1 ft². \(1\,\mathrm{ft^2} = 144\,\mathrm{in^2}\). |
Terms
| rectangular prism | A box shape whose 6 faces are all rectangles (some may be squares). Tissue boxes, bricks and fish tanks are all rectangular prisms, the most common solid around us. |
| surface area | The total area of the outside of a solid. For a rectangular prism, it is the sum of the areas of all 6 faces. It tells you how much you need to wrap, paint or cover something. |
| net | The flat shape you get by cutting a solid open and laying it out. The net of a rectangular prism is made of 6 rectangles, and their total area is the surface area. |
| area of the base | The area of the bottom face of a solid. For a rectangular prism, it is the area of the bottom rectangle (length × width). For surface area it is doubled to cover both the bottom and the top. |
| lateral area | The total area of the side faces of a solid. For a prism, the sides unfold into one rectangle, so the lateral area is "perimeter of the base × height". |
| prism | A solid made by stacking a polygon straight up without changing its shape. A rectangular prism is a prism whose base is a rectangle. |
| unit conversion | Writing the same amount in a different unit. For area, the key point is that the area factor is the length factor squared (1 ft = 12 in, so 1 ft² = 144 in²). |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Mixed multiplication and addition (Grades 3–5) |
|
| Area of a rectangle (Grades 3–4) |
|
| Rectangular prisms, cubes and nets (Grades 5–6) |
|
| Surface area of prisms (Grades 6–7) |
|
| Multiplying decimals (Grades 5–6) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the surface area of a rectangular prism
| Length (in) | 20 |
| Width (in) | 30 |
| Height (in) | 10 |
| Surface area (in²) | =(B1*B2+B1*B3+B2*B3)*2 |
Table to find it as "area of the base × 2 + lateral area"
| Length (in) | 20 |
| Width (in) | 30 |
| Height (in) | 10 |
| Area of the base (in²) | =B1*B2 |
| Perimeter of the base (in) | =(B1+B2)*2 |
| Lateral area (in²) | =B5*B3 |
| Surface area (in²) | =B4*2+B6 |
Table to convert square inches to square feet
| Surface area in in² | 2200 |
| Surface area in ft² | =B1/144 |
After pasting, column A holds the labels and column B holds the numbers. The upper rows are your inputs, and the formulas in the lower rows calculate automatically from them.
In a formula, "B1" and "B2" mean "use the number in that cell", "*" is multiplication, "/" is division and "+" is addition.
In the first table, for example, B4 shows (20 × 30 + 20 × 10 + 30 × 10) × 2 = 2200 (in², since the inputs are in inches). The second table shows the same 2200 in B7, and the third table shows about 15.28 (ft²) in B2. Just replace the input numbers with your own lengths.
In a formula, "B1" and "B2" mean "use the number in that cell", "*" is multiplication, "/" is division and "+" is addition.
In the first table, for example, B4 shows (20 × 30 + 20 × 10 + 30 × 10) × 2 = 2200 (in², since the inputs are in inches). The second table shows the same 2200 in B7, and the third table shows about 15.28 (ft²) in B2. Just replace the input numbers with your own lengths.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the surface area of a rectangular prism
| Length (in) | 20 |
| Width (in) | 30 |
| Height (in) | 10 |
| Surface area (in²) | =(B1*B2+B1*B3+B2*B3)*2 |
Table to find it as "area of the base × 2 + lateral area"
| Length (in) | 20 |
| Width (in) | 30 |
| Height (in) | 10 |
| Area of the base (in²) | =B1*B2 |
| Perimeter of the base (in) | =(B1+B2)*2 |
| Lateral area (in²) | =B5*B3 |
| Surface area (in²) | =B4*2+B6 |
Table to convert square inches to square feet
| Surface area in in² | 2200 |
| Surface area in ft² | =B1/144 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the input numbers with your own lengths.
How to calculate it in Python
length = 20 # length (inches in this example)
width = 30 # width (same unit as length)
height = 10 # height (same unit as length)
# surface area (3 kinds of face areas added up, times 2; in2 in this example)
area = (length * width + length * height + width * height) * 2
area_ft2 = area / 144 # in square feet (for inputs in inches)
print(f"Surface area: {area} in2")
print(f"In square feet: {area_ft2} ft2")
Runs with the standard library only. "*" is multiplication, "/" is division and "+" is addition. Change the length, width and height at the top and run it. This example uses inches; if you use feet, the area is in square feet, and multiplying it by 144 gives square inches.
How to write it in LaTeX and other math languages (copy and paste)
Surface area of a rectangular prism
S = (l × w + l × h + w × h) × 2
S = (l \times w + l \times h + w \times h) \times 2
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>l</mi>
<mo>×</mo>
<mi>w</mi>
<mo>+</mo>
<mi>l</mi>
<mo>×</mo>
<mi>h</mi>
<mo>+</mo>
<mi>w</mi>
<mo>×</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mn>2</mn>
</mrow>
</math>
S = (l xx w + l xx h + w xx h) xx 2
2*(l*w + l*h + w*h)
S := 2*(l*w + l*h + w*h);
S = 2*(l*w + l*h + w*h);
S = (l × w + l × h + w × h) × 2
Base area × 2 + lateral area (surface area of a prism)
S = S₁ × 2 + L × h
S = S_1 \times 2 + L \times h
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<msub><mi>S</mi><mn>1</mn></msub>
<mo>×</mo>
<mn>2</mn>
<mo>+</mo>
<mi>L</mi>
<mo>×</mo>
<mi>h</mi>
</mrow>
</math>
S = S_1 xx 2 + L xx h
s1*2 + L*h
S := S1*2 + L*h;
S = S1*2 + L*h;
S = S_1 × 2 + L × h
Converting area units (in² to ft²)
S[ft²] = S[in²] ÷ 144
S_{\mathrm{ft^2}} = S_{\mathrm{in^2}} \div 144
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>S</mi><mrow><msup><mi mathvariant="normal">ft</mi><mn>2</mn></msup></mrow></msub>
<mo>=</mo>
<msub><mi>S</mi><mrow><msup><mi mathvariant="normal">in</mi><mn>2</mn></msup></mrow></msub>
<mo>÷</mo>
<mn>144</mn>
</mrow>
</math>
S_(ft^2) = S_(in^2) -: 144
sIn2/144
sFt2 := sIn2/144;
s_ft2 = s_in2/144;
S(ft²) = S(in²)/144
How to have ChatGPT do the calculation
You are a math calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). A box shaped like a rectangular prism is 20 inches long, 30 inches wide and 10 inches tall. Find each of the following: 1. The surface area of the box in square inches (in²) 2. That surface area in square feet (1 ft² = 144 in²) Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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