Cuboid surface area from a net

Cuboid surface area from a net

For volume we counted the space inside a solid. For surface area we count how much area its faces take up together. In other words: how much wrapping paper would you need to cover the cuboid completely?

A net helps you compute surface area

A cuboid net is 6 rectangles laid flat. When you compute the area of all 6 rectangles and add them up, you get the surface area.

In a cuboid the faces always come in equal pairs:

  • front = back (rectangle a × c)
  • left = right (b × c)
  • top = bottom (a × b)

So you don't need to compute six different areas – just three different pairs and double each.

Formula

S = 2 · (a · b + a · c + b · c)

Where:

  • a, b, c are the lengths of the three edges of the cuboid
  • a · b is the area of one pair of faces (top = bottom)
  • a · c is the area of another pair (front = back)
  • b · c is the area of the third pair (left = right)

Add the three areas in the parentheses and double the result (because every pair is the same size).

Worked example

A cuboid measures 5 cm by 4 cm by 3 cm. What is its surface area?

Solution:

  • a · b = 5 · 4 = 20 cm²
  • a · c = 5 · 3 = 15 cm²
  • b · c = 4 · 3 = 12 cm²

Add: 20 + 15 + 12 = 47

Double: 2 · 47 = 94 cm²

The surface area of the cuboid is 94 cm².

Surface area of a cube

For a cube all faces are equal (because a = b = c). The formula simplifies:

S = 6 · a · a

Six faces, each with area a · a.

Example: a cube with edge 5 cm has S = 6 · 5 · 5 = 6 · 25 = 150 cm².

Watch the units

  • Surface area is in square units (cm², m², dm²) – because it is an area (2D), not a space.
  • Volume is in cubic units (cm³, m³, dm³).
  • They are not the same! If someone tells you that a cuboid has a surface area of 1 m³ – that is nonsense. m³ is a volume.

A check

When you have a net in front of you (on paper or in an exercise), you can verify your answer like this:

  • Compute the area of each of the 6 rectangles separately.
  • Add them up.
  • The total should match the result from S = 2 · (a · b + a · c + b · c).

If they don't match, somewhere you made a mistake – either in addition or you swapped dimensions.

Practice