How to Find the Surface Area of a Rectangular Prism
Read,3 minutes
A rectangular prism has six rectangular faces. Surface area is the total area of all outside faces, so it is measured in square units.
Here is an example of a rectangular prism:
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Surface Area Formula
For length \(l\), width \(w\), and height \(h\), the surface area is:
\(SA = 2(lw + lh + wh)\)
This works because a rectangular prism has two faces of each type: top and bottom, front and back, and left and right.
Example
If a prism is \(5\) cm by \(4\) cm by \(3\) cm, then \(SA = 2(5 \cdot 4 + 5 \cdot 3 + 4 \cdot 3) = 94\).
Surface Area of a Rectangle Prism
Think of this lesson as more than a rule to memorize. Surface Area of a Rectangle Prism is about three-dimensional measurement, volume, and surface area. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.
Geometry formulas work because they measure a specific feature: length around, space inside, or space enclosed by a solid. Match the question to the measurement first.
Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.
- Sketch or label the shape.
- Decide whether the question asks for length, area, volume, or surface area.
- Substitute values into the matching formula.
- Keep units squared for area and cubed for volume.
A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.
Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.
When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.
On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.
Free printable Worksheets
Exercises for Surface Area of a Rectangle Prism
1) A rectangular prism has dimensions \(2\), \(3\), and \(4\) units. Find its surface area.
2) A rectangular prism has dimensions \(5\), \(6\), and \(7\) cm. Find its surface area.
3) A rectangular prism has dimensions \(8\), \(4\), and \(3\) units. Find its surface area.
4) A rectangular prism has dimensions \(10\), \(9\), and \(2\) in. Find its surface area.
5) A rectangular prism has dimensions \(12\), \(5\), and \(4\) units. Find its surface area.
6) A rectangular prism has dimensions \(15\), \(10\), and \(6\) ft. Find its surface area.
7) A cube has side length \(5\). Find its surface area.
8) A rectangular prism has dimensions \(\frac{5}{2}\), \(4\), and \(6\) m. Find its surface area.
9) An open-top box is \(8\) by \(5\) by \(3\). Find the outside area excluding the top.
10) A prism has surface area \(208\), length \(8\), and width \(6\). Find the height.
11) Prism A is \(6\times5\times4\); Prism B is \(7\times4\times3\). Which has greater surface area and by how much?
12) A rectangular prism has dimensions \(14\), \(10\), and \(6\) in. Find its surface area.
13) A \(12\times8\times5\) prism is painted on the sides only, excluding top and bottom. Find the lateral area.
14) Total edge length is \(88\). Length is \(10\) and width is \(7\). Find the surface area.
15) A prism with surface area \(50\) has every dimension multiplied by \(3\). Find the new surface area.
16) An open-top tank is \(20\) by \(12\) by \(10\). Find the material area.
17) A prism has dimensions \(x\), \(x + 2\), and \(5\); surface area is \(236\). Find \(x\).
18) A prism is \(3\) by \(4\) by \(h\) and has surface area \(94\). Find \(h\).
19) A \(18\times12\times9\) box has a label covering \(100\) square units. How much surface remains visible?
20) Which has greater surface area: \(4\times5\times6\) or \(3\times5\times8\)? By how much?
1) Use \(SA = 2(lw + lh + wh)\).
Substitute: \(SA = 2(6 + 8 + 12)\).
Compute: \(SA = 52\).
Answer: \(\color{red}{52\ \text{units}^2}\)
2) Use \(SA = 2(lw + lh + wh)\).
Substitute: \(SA = 2(30 + 35 + 42)\).
Compute: \(SA = 214\).
Answer: \(\color{red}{214\ \text{cm}^2}\)
3) Use \(SA = 2(lw + lh + wh)\).
Substitute: \(SA = 2(32 + 24 + 12)\).
Compute: \(SA = 136\).
Answer: \(\color{red}{136\ \text{units}^2}\)
4) Use \(SA = 2(lw + lh + wh)\).
Substitute: \(SA = 2(90 + 20 + 18)\).
Compute: \(SA = 256\).
Answer: \(\color{red}{256\ \text{in}^2}\)
5) Use \(SA = 2(lw + lh + wh)\).
Substitute: \(SA = 2(60 + 48 + 20)\).
Compute: \(SA = 256\).
Answer: \(\color{red}{256\ \text{units}^2}\)
6) Use \(SA = 2(lw + lh + wh)\).
Substitute: \(SA = 2(150 + 90 + 60)\).
Compute: \(SA = 600\).
Answer: \(\color{red}{600\ \text{ft}^2}\)
7) Use \(SA = 6s^2\).
Compute \(6(5^2) = 150\).
Answer: \(\color{red}{150\ \text{square units}}\)
8) Use \(SA = 2(lw + lh + wh)\).
Substitute: \(SA = 2(10 + 15 + 24)\).
Compute: \(SA = 98\).
Answer: \(\color{red}{98\ \text{m}^2}\)
9) Bottom: \(8\cdot5 = 40\).
Sides: \(2(8\cdot3) + 2(5\cdot3) = 78\).
Total: \(118\).
Answer: \(\color{red}{118\ \text{square units}}\)
10) Set \(208 = 2(48 + 8h + 6h)\).
Simplify \(208 = 96 + 28h\).
Solve \(h = 4\).
Answer: \(\color{red}{4\ \text{units}}\)
11) A: \(2(30 + 24 + 20) = 148\).
B: \(2(28 + 21 + 12) = 122\).
A is greater by \(26\).
Answer: \(\color{red}{\text{Prism A by }26}\)
12) Use \(SA = 2(lw + lh + wh)\).
Substitute: \(SA = 2(140 + 84 + 60)\).
Compute: \(SA = 568\).
Answer: \(\color{red}{568\ \text{in}^2}\)
13) Lateral area is \(2lh + 2wh\).
Compute \(2(12\cdot5)+2(8\cdot5)=200\).
Answer: \(\color{red}{200\ \text{square units}}\)
14) Use \(4(l+w+h)=88\), so \(h=5\).
Surface area: \(2(70+50+35)=310\).
Answer: \(\color{red}{310\ \text{square units}}\)
15) Surface area scales by \(3^2 = 9\).
New area: \(50\cdot9 = 450\).
Answer: \(\color{red}{450\ \text{square units}}\)
16) Bottom: \(20\cdot12 = 240\).
Sides: \(2(20\cdot10)+2(12\cdot10)=640\).
Total: \(880\).
Answer: \(\color{red}{880\ \text{square units}}\)
17) Set \(236 = 2(x(x+2)+5x+5(x+2))\).
Simplify to \(x^2 + 12x - 108 = 0\).
The positive solution is \(x = 6\).
Answer: \(\color{red}{6}\)
18) Set \(94 = 2(12+3h+4h)\).
Simplify \(94 = 24 + 14h\).
Solve \(h = 5\).
Answer: \(\color{red}{5}\)
19) Total surface area: \(2(216+162+108)=972\).
Visible area: \(972-100=872\).
Answer: \(\color{red}{872\ \text{square units}}\)
20) First: \(2(20+24+30)=148\).
Second: \(2(15+24+40)=158\).
Second is greater by \(10\).
Answer: \(\color{red}{\text{the }3\times5\times8\text{ prism by }10}\)
Surface Area of a Rectangle Prism Practice Quiz
