Volume of an L-Shaped Prism: Step-by-Step

Updated 6 November 2026 · Standard formula, shown below

This guide shows you how to calculate the volume of an L-shaped prism (also called a compound prism). You’ll need the prism’s height and the dimensions of its L-shaped cross-section. By the end, you’ll be able to find the volume in cubic units (like cm³ or in³) using a simple two-step method: find the area of the L-shape, then multiply by the prism’s length.

What You’ll Need

A ruler, tape measure, or given dimensions for the L-shaped cross-section
The length (or height) of the prism
A calculator (optional but helpful)
Paper and pencil to sketch the shape and label dimensions

Step-by-Step

Identify the cross-section. The L-shaped prism has a uniform cross-section along its length. That cross-section is an L-shape, which you can think of as two rectangles joined together.
Sketch the L-shape and label all outer dimensions. You need the total height, total width, and the thickness of the arms. For example, a typical L has a total height H, total width W, and both arms have thickness t (they may differ, so label each).
Divide the L-shape into two rectangles. Draw a line to split it. Common ways:
- Split horizontally: top rectangle has width W and height t; bottom rectangle has width t and height H – t.
- Or split vertically: left rectangle has height H and width t; right rectangle has height H – t and width W – t.
Choose whichever is easier with your given dimensions.
Calculate the area of each rectangle. Use the formula: area = length × width. For the split described above:
- Top rectangle area = W × t
- Bottom rectangle area = t × (H – t)
Add the two areas to get the total cross-sectional area. Area of L-shape = (W × t) + (t × (H – t)). Simplify if you like: = t × (W + H – t).
Measure the prism’s length. This is the dimension perpendicular to the cross-section. It is often called the height of the prism, but to avoid confusion, call it length L.
Multiply the cross-sectional area by the length. Volume = Area of L-shape × L. That’s your answer in cubic units.
Check units. If your dimensions are in centimeters, volume is in cm³; if in inches, in³; if in feet, ft³, etc. Make sure all measurements are in the same unit before multiplying.

Common Mistakes

Using the outer dimensions as if the L were a solid rectangle. That would give the volume of a rectangular prism, not an L. Always subtract the missing part.
Forgetting to subtract the thickness when finding the second rectangle’s dimensions. For example, the bottom rectangle’s height is not H, but H – t.
Mixing units. If one dimension is in meters and another in centimeters, convert everything to the same unit first.
Confusing the prism’s length with the cross-section’s height. The length is the dimension that runs the whole way through; the cross-section is the face you see when you look at the end.
Not simplifying the area formula correctly. Double-check your algebra: t × (W + H – t) is correct only if both arms have the same thickness t. If they differ, use separate thicknesses.

Tips

Draw a clear diagram. Label every side, even if you think it’s obvious. This prevents mix-ups.
Use the “missing rectangle� method. Instead of splitting, you can calculate the area of the larger rectangle (W × H) and subtract the area of the missing corner ( (W – t) × (H – t) ). This often has fewer steps.
Write down each step. It’s easier to catch errors when you see the numbers.
Use a calculator for the final multiplication. Especially with decimals, manual multiplication can introduce errors.
Check your answer’s reasonableness. The volume should be less than the volume of a rectangular prism with the same outer dimensions. If it’s larger, you probably forgot to subtract the missing part.

Troubleshooting

Your volume seems too large. Likely you used the full outer rectangle area. Subtract the missing corner or re-split correctly.
Your volume seems too small. Check that you used the correct thickness. If the thickness is small, the area will be small, but make sure you didn’t subtract twice.
You have different thicknesses for the two arms. Then you can’t use the simplified formula. Split into two rectangles with their own dimensions: one rectangle is width × thickness1, the other is (height – thickness1) × thickness2 (or similar, depending on the split).
Your dimensions are in different units. Convert all to the same unit before calculating. For example, if you have meters and centimeters, convert meters to centimeters (multiply by 100) or vice versa.
You’re not sure which dimension is the length. The length is the one that is the same for every cross-section along the prism. If you have a physical object, it’s the dimension you can measure along the “extrusion� direction.
Your L-shape is not a right angle. If the arms are not perpendicular, the area calculation becomes more complex (you might need trigonometry). This method assumes right angles, which is typical for an L-shaped prism.

Frequently Asked Questions

Can I use this method for any L-shaped prism?

Yes, as long as the cross-section is a simple L-shape with right angles and the prism has a uniform cross-section along its length. If the arms have different thicknesses, just split accordingly.

What if the L-shape has arms of different thicknesses?

Then you cannot use the simplified formula t × (W + H – t). Instead, split into two rectangles with their own dimensions. For example, if the vertical arm has thickness a and the horizontal arm has thickness b, then the area = a × H + b × (W – a) (if you split that way).

Do I need to convert units before calculating?

Yes. All dimensions must be in the same unit. If you mix meters and centimeters, convert everything to one unit first. The volume will be in cubic units of that unit.

How do I find the volume if I only have the outer dimensions and the thickness?

That’s exactly what this method does. Use the missing rectangle method: area = W × H – (W – t) × (H – t). Then multiply by the length.

What is the difference between the height of the cross-section and the height of the prism?

The cross-section height is one dimension of the L-shape (the vertical part). The prism’s height (or length) is the dimension that runs perpendicular to the cross-section. In many problems, the prism’s height is called the length. Always check which is which.

Can I find the volume if the L-shape is not drawn to scale?

Yes, you just need the numeric dimensions. The drawing’s scale doesn’t matter; only the labeled measurements do.

Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.