Key takeaway: Surface area adds up every outer face of a solid to say how much covering it needs; for a box, compute three face sizes, add them, and double the result.
Grade 66.G.A.4Geometry and Measurement
Definition of Surface Area
The surface area of a solid tells how much material it takes to cover the whole outside, the way wrapping paper covers a gift. It is counted in square units because each face is a flat region.
To find it, list every face, work out how big each one is, and add them all. A box has faces in three matching pairs, so only three sizes are needed before doubling. This quantity differs from volume: one measures the skin of the solid, the other the space inside. An unfolded pattern of the solid, called a net, lays every face out together and makes the adding easy to follow.
Formula and notation
For a box with length \(\ell\), width \(w\) and height \(h\), the three pairs of faces have areas \(\ell w\), \(\ell h\) and \(w h\):
\[SA = 2(\ell w + \ell h + w h)\]
The result is in square units, such as \(\text{in}^2\) or \(\text{cm}^2\).
Key facts
- Six faces, in three equal pairs, make up the covering of a box.
- The unit is square inches, square feet or similar, never cubic ones.
- Painting or wrapping a solid depends on its surface area, not on the space inside.
- Adding the faces in a net gives the same surface area as the formula.
Surface Area example
A gift box measures 12 inches long, 5 inches wide and 3.5 inches tall. How much wrapping paper covers the whole box with no overlap?
-
Three face areas: \(12 \times 5 = 60\), \(12 \times 3.5 = 42\), \(5 \times 3.5 = 17.5\).
-
Add them: \(60 + 42 + 17.5 = 119.5\).
-
Double for the opposite faces: \(2 \times 119.5 = 239\).
The box needs 239 square inches of paper.
Common mistake
Watch out
Wren adds only the three faces she can see in a picture and answers 119.5 square inches.
Why it is wrong
A box has six faces. The three hidden ones match the three visible ones, so the total must be doubled: \(2 \times 119.5 = 239\). Check that the number of faces you added is six.
Related terms and lessons
- Grade 6 · Surface Area and NetsLessonWorksheetQuiz
- Grade 7 · Area, Surface Area and Volume of PrismsLessonWorksheetQuiz
Browse the glossary: A–Z list · Grade 6 vocabulary · Geometry and Measurement