Math101 · Independent Ontario learning library
Math101learn.math101.caSurface Area
Surface area measures the total exposed area of a three-dimensional object by accounting for every face and curved surface exactly once.
Surface area is the total area of every exposed face and curved surface of a three-dimensional object.
Think in nets
A net unfolds a solid into flat regions. Surface area becomes an area-accounting problem: identify the shapes, calculate each area, and add each exposed piece once.
This perspective is safer than memorizing disconnected formulas. It also handles open containers and composite objects, where some faces are missing or hidden.
Rectangular prisms
A rectangular prism has three pairs of congruent faces:
An open-top box would omit one $lw$ face, giving $158-40=118$ cm².
Prisms in general
For any right prism,
where $B$ is the area of one base, $P$ is its perimeter, and $h$ is prism length or height. The lateral faces unfold into a rectangle with dimensions $P$ and $h$.
This formula works for triangular and polygonal prisms when the base measurements are known.
Cylinders
A closed cylinder has two circular bases and one rectangular lateral surface:
The lateral rectangle has one side $h$ and the other equal to the circle’s circumference $2\pi r$. For a label around a can, only lateral area $2\pi rh$ is needed.
Pyramids and cones
A regular pyramid has base area $B$ and triangular lateral faces. If base perimeter is $P$ and slant height is $s$,
A cone similarly has
Slant height lies along the surface; vertical height runs through the interior. They are not interchangeable.
Spheres
A sphere of radius $r$ has surface area
This is four times the area of a circle with the same radius. Diameter must be halved before substitution.
Composite objects
Break the object into known solids, but do not count surfaces where pieces touch. Add only visible exterior areas. For a cylinder attached to a rectangular base, the shared circular region is hidden on both components and must be excluded appropriately.
Scale and units
Surface area uses square units. A length scale factor $k$ produces surface-area factor $k^2$. If every dimension triples, surface area becomes $9$ times as large.
Common mistakes
Counting a face twice or missing it. Label a net or checklist every exposed surface.
Using vertical height for cone or pyramid lateral area. Use slant height.
Including hidden contact surfaces. Composite solids expose only the outside.
Reporting cubic units. Surface area is two-dimensional and uses square units.
Quick self-check
- Which faces or curved surfaces are exposed?
- Is the container open or closed?
- Have I distinguished radius, vertical height, and slant height?
- Are hidden joins removed and units squared?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the surface area of a closed rectangular prism measuring 8 cm by 5 cm by 3 cm.
- SA = 2(8)(5) + 2(8)(3) + 2(5)(3)
- = 80 + 48 + 30
- = 158 cm²
End of lesson
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