Surface Area of Prisms and Cylinders
Calculate surface area of rectangular and triangular prisms and cylinders using nets. Apply to real-world problems involving packaging, painting, and material costs.
A lesson of Pre-Algebra, in Geometry Foundations, about 10 minutes.
Surface Area
By the end of this lesson, you will be able to find the surface area of prisms, pyramids, and cylinders by adding the areas of all faces.
What Is Surface Area?
Surface area (SA): the total area of all outer surfaces of a 3D figure. Measured in square units.
Strategy: unfold the figure into a net (flat layout of all faces), find each face's area, then add them.
Rectangular Prism (Box)
6 faces: 3 pairs of identical rectangles.
$$SA = 2lw + 2lh + 2wh$$
Example: $l=8$, $w=5$, $h=3$
$SA = 2(40) + 2(24) + 2(15) = 80 + 48 + 30 = 158$ square units
Cube
All 6 faces are equal squares.
$$SA = 6s^2$$
Example: $s = 4$ cm
$SA = 6(16) = 96$ cm²
Triangular Prism
2 triangular bases + 3 rectangular lateral faces.
$SA = 2 \times (\text{triangle area}) + (\text{sum of 3 rectangular faces})$
Cylinder
2 circular bases + 1 lateral surface (rectangle when unrolled, width = circumference).
$$SA = 2\pi r^2 + 2\pi rh$$
Example: $r = 3$, $h = 10$
$SA = 2\pi(9) + 2\pi(3)(10) = 18\pi + 60\pi = 78\pi \approx 245.0$ square units
Summary
SA = sum of all face areas.
Box: $2lw + 2lh + 2wh$. Cube: $6s^2$. Cylinder: $2\pi r^2 + 2\pi rh$.
Net strategy: unfold the figure, find each face, add them all.