Enter Cuboid Dimensions
Enter positive dimensions. Length, width, and height must use the same unit.
Calculation Steps
The steps update automatically when you change a dimension or unit.
Enter dimensions to see the calculation steps.
Cuboid Surface Area Formulas
A cuboid, also known as a rectangular prism, is a three-dimensional solid with six rectangular faces. Opposite faces have equal areas. Its dimensions are length, width, and height.
SA = 2(lw + lh + wh)
The total surface area is the sum of all six faces: two length-by-width faces, two length-by-height faces, and two width-by-height faces.
LSA = 2h(l + w)
Lateral surface area includes the four side faces and excludes the top and bottom. This formula assumes the base is the rectangle with dimensions l × w and the vertical height is h.
Atop+bottom = 2lw
Length × width: A = lw
Length × height: A = lh
Width × height: A = wh
V = lwh
Here, l is length, w is width, and h is height. Use the same unit for all three dimensions. Surface area is expressed in square units, while volume is expressed in cubic units.
Worked Examples
Example 1: Cuboid with dimensions 10 cm × 5 cm × 4 cm
Given: l = 10 cm, w = 5 cm, h = 4 cm.
SA = 2(lw + lh + wh)
SA = 2((10 × 5) + (10 × 4) + (5 × 4))
SA = 2(50 + 40 + 20) = 220 cm².
Lateral area = 2 × 4 × (10 + 5) = 120 cm².
Volume = 10 × 5 × 4 = 200 cm³.
Example 2: Box measuring 20 cm × 12 cm × 8 cm
Given: l = 20 cm, w = 12 cm, h = 8 cm.
SA = 2((20 × 12) + (20 × 8) + (12 × 8))
SA = 2(240 + 160 + 96) = 992 cm².
Lateral area = 2 × 8 × (20 + 12) = 512 cm².
Volume = 20 × 12 × 8 = 1,920 cm³.
Example 3: Cuboid measuring 2 m × 1.5 m × 1 m
Given: l = 2 m, w = 1.5 m, h = 1 m.
SA = 2((2 × 1.5) + (2 × 1) + (1.5 × 1))
SA = 2(3 + 2 + 1.5) = 13 m².
Volume = 2 × 1.5 × 1 = 3 m³.
Surface Area and Volume Units
Surface area converts by the square of a length conversion factor. Volume converts by the cube of that factor. For example, 1 m = 100 cm, but 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³.
| Measurement | Equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 m² | 1,000,000 mm² |
| 1 ft² | 144 in² |
| 1 yd² | 9 ft² |
| 1 m³ | 1,000,000 cm³ |
| 1 ft³ | 1,728 in³ |
How to Calculate the Surface Area of a Cuboid
- Measure or enter the cuboid's length, width, and height.
- Choose the unit used for all three dimensions.
- Select the unit for the surface area and volume results.
- Choose which surface-area value you want highlighted.
- Select Calculate Surface Area to view the results and calculation steps.
What is the difference between surface area and volume?
Surface area measures the total area covering the outside of a solid. It is useful when estimating paint, wrapping paper, labels, or material needed to cover a box. Volume measures the space inside the solid and is useful for capacity and storage calculations. The amount of material needed for a real object may differ because of overlap, thickness, openings, and waste.
Where is the cuboid formula used?
Cuboid surface-area calculations are useful for cartons, storage boxes, rooms, rectangular tanks, packaging, building materials, and classroom geometry. For a closed rectangular box, total surface area includes all six faces. For an open-top box, the top face should be excluded from the calculation.
Frequently Asked Questions
1. What is the formula for the surface area of a cuboid?
The formula is SA = 2(lw + lh + wh). Multiply each pair of dimensions, add the three products, and multiply the sum by two.
2. How do you calculate lateral surface area?
Use LSA = 2h(l + w). This gives the combined area of the four side faces, excluding the top and bottom.
3. How do you calculate the volume of a cuboid?
Multiply length by width by height: V = l × w × h. The result is expressed in cubic units.
4. Is a cuboid the same as a rectangular prism?
Yes. The terms are commonly used for a solid with six rectangular faces, twelve edges, and eight vertices.
5. Can the dimensions use different units?
You should convert the dimensions to a common unit before applying the formula. This calculator uses one selected unit for length, width, and height, then converts the calculated area and volume into your chosen output units.
6. What is the surface area of a cuboid with length 10 cm, width 5 cm, and height 4 cm?
SA = 2((10 × 5) + (10 × 4) + (5 × 4)) = 220 cm².
7. Does total surface area include the top and bottom?
Yes. Total surface area includes all six faces. Lateral surface area includes only the four side faces and excludes the top and bottom.
8. How do I calculate the area of an open-top cuboid?
If the top face is completely absent, subtract the top face area lw from the total surface area. The area of the remaining five faces is SA = lw + 2lh + 2wh.