A rectangular prism’s total surface area is 2(lw + lh + wh) and its lateral area is 2h(l +.
You probably remember that a box has length, width, and height. But when a test or a DIY project asks for “the area,” it’s easy to pause. Do you need to cover the whole thing, or just the sides? That’s the difference between total surface area and lateral surface area.
The area formula for a rectangular prism isn’t just one formula—it’s two related ones. The total surface area (TSA) covers all six faces. The lateral surface area (LSA) covers only the four side faces. Both start with the same three measurements: length (l), width (w), and height (h). Here is exactly how each one works, when to use it, and why it matters.
Surface Area vs. Lateral Surface Area
A rectangular prism (also called a cuboid) is a solid with six flat rectangular faces. Opposite faces are identical. This shape is everywhere—shipping boxes, bricks, rooms, and books. Because it’s three-dimensional, you can’t just multiply length by width. You need to account for every surface.
The total surface area adds up every visible side: top, bottom, front, back, left, and right. The lateral surface area ignores the top and bottom. It covers only the sides. That makes LSA useful for tasks like painting the walls of a room where you don’t paint the floor and ceiling.
The units are always square units—square inches, square feet, or square centimeters. If you mix inches and feet, the formula won’t give you a reliable answer. Consistent measurements are essential for an accurate result.
Why Are There Two Formulas?
The existence of two formulas isn’t meant to confuse you. It’s about matching the math to the real-world task. Choosing the wrong one means buying too much wrapping paper or running out of paint halfway through the job.
- Total Surface Area (TSA): Use this when you need to cover the entire object. Wrapping a gift, calculating the material for a box, or finding the surface area of a brick all call for TSA.
- Lateral Surface Area (LSA): Use this when the top and bottom are excluded. Painting the walls of a room, covering the sides of a column, or wrapping a label around a can (though a can is a cylinder, the concept is similar).
- The Formula Relationship: LSA is always a subset of TSA. You can find TSA by adding the base areas (2lw) to the LSA. Mathematically, TSA = LSA + 2lw.
- Memory Trick: Think “Lateral = Sides.” Total = All. The expanded formula TSA = 2lw + 2lh + 2wh explicitly shows you three pairs of identical rectangles.
This distinction matters on standardized tests too. A question might specifically ask for the “lateral surface area” to see if you know to exclude the bases. Reading the prompt carefully is half the battle.
Breaking Down the TSA Formula
The standard total surface area formula is TSA = 2(lw + lh + wh). It’s compact, but it helps to expand it: SA = 2lw + 2lh + 2wh. Here, lw is the area of the top (or bottom), lh is the front (or back), and wh is the side (left or right).
The Andrews website defines the rectangular prism’s base as the top and bottom faces. Since they are congruent, calculating 2lw efficiently captures both. This pairing is what makes the formula elegant—you only need three multiplications to cover all six faces.
Let’s run a quick example. A box with l=4, w=3, h=2. First, lw = 12. lh = 8. wh = 6. Sum them: 12 + 8 + 6 = 26. Multiply by 2. TSA = 52 square units. That’s the total wrapping paper needed for the entire box.
| Quantity | Formula | What It Covers |
|---|---|---|
| Total Surface Area | 2(lw + lh + wh) | All six faces |
| Lateral Surface Area | 2h(l + w) | Four side faces |
| Volume | l × w × h | Interior space |
| Base Perimeter | 2(l + w) | Perimeter of one base |
| Space Diagonal | √(l² + w² + h²) | Longest diagonal inside |
The space diagonal formula is useful for finding the longest distance between two opposite corners of the prism. It’s the 3D version of the Pythagorean theorem and comes in handy for packing or construction problems.
How to Calculate It Step-by-Step
Memorizing the formula is one thing. Applying it without panic is another. Here is a reliable step-by-step process for any rectangular prism problem.
- Identify and Label l, w, and h: Know which dimension is which. Length is usually the longest side, width is the shorter base side, and height is the vertical measurement. Label them on the diagram.
- Decide: TSA or LSA? Read the problem to know if you need the entire surface or just the sides. Key words for LSA include “lateral,” “sides,” and “excluding the top and bottom.”
- Plug into the Correct Formula: Write down the formula. TSA = 2(lw + lh + wh). Substitute the numbers for the variables. Keep the units (inches, feet, cm) in your work to avoid mix-ups.
- Apply the Order of Operations: Do the multiplications inside the parentheses first (lw, lh, wh). Add those three products together. Finally, multiply by 2. Double-check your arithmetic before moving on.
A common mistake is calculating l × w × h (volume) instead of surface area. Remember, volume uses cubic units, while surface area uses square units. Writing the correct unit exponent (² vs ³) is a great way to double-check your intent.
What About a Square Prism?
A special rectangular prism is one where the length and width are equal. This is simply a prism with a square base (like a cube, though a cube has all sides equal). Per the rectangular prism definition on Cuemath, this is still a rectangular prism, just a specific type.
When l = w, the TSA formula simplifies. If l = w = s (where s is the side of the square base), then TSA = 2(s² + sh + sh) = 2(s² + 2sh) = 2s² + 4sh. This is the classic formula for a square prism.
The lateral surface area also simplifies. LSA = 2h(l + w) becomes LSA = 2h(2s) = 4sh. This makes sense—you have four identical rectangular side faces, each with area s × h. Recognizing this pattern can save you time on tests.
| Feature | General Rectangular Prism | Square Prism (l = w) |
|---|---|---|
| TSA | 2(lw + lh + wh) | 2s² + 4sh |
| LSA | 2h(l + w) | 4sh |
| Base Condition | l ≠ w (generally) | l = w = s |
The Bottom Line
The “area” of a rectangular prism usually means its surface area. Keep the two formulas straight: TSA = 2(lw + lh + wh) for the whole box, and LSA = 2h(l + w) for just the sides. Pick the right one based on what the problem asks or the project demands.
If you’re working through a geometry unit and these formulas feel tricky to apply, your math teacher or a tutor can walk you through the steps using your specific homework problems and diagrams.
References & Sources
- Andrews. “Rectangular Prism Base Definition” The “base” of a rectangular prism is typically considered the top and bottom faces, which are congruent rectangles.
- Cuemath. “Surface Area of Rectangular Prism” A rectangular prism is a three-dimensional solid shape with six rectangular faces, also known as a cuboid.