Surface Area of Prisms and Cylinders
Calculate lateral area and total surface area of prisms and cylinders using nets and formulas. Solve design and cost problems that require minimizing or computing surface area.
A lesson of Geometry, in Area and Volume, about 10 minutes.
By the end of this lesson, you will be able to find the total and lateral surface areas of right prisms and right cylinders, use nets to visualize surface area, and apply these formulas to real-world contexts like packaging and painting.
The Big Idea
Surface area is the total area of all faces of a three-dimensional solid. A net unfolds the solid into a flat shape, making it easy to see all faces. For prisms and cylinders, the lateral (side) area can be computed as the perimeter of the base times the height — a formula that works for any right prism.
Prisms
A prism has two parallel, congruent polygonal bases connected by rectangular lateral faces.
Right prism: the lateral edges are perpendicular to the bases (the sides are rectangles).
Lateral surface area: $LA = Ph$ (perimeter of base × height)
Total surface area: $SA = LA + 2B = Ph + 2B$
where $B$ = area of one base.
Cylinders
A cylinder is like a prism with circular bases.
Lateral surface area: When the cylinder is "unrolled," the lateral face becomes a rectangle with width = circumference of base and height = $h$.
$$LA = 2\pi r h$$
Total surface area:
$$SA = 2\pi r h + 2\pi r^2 = 2\pi r(h + r)$$
Example 1 — Rectangular Prism (Box)
Box: length 8, width 5, height 3.
$B = 8 \times 5 = 40$
$P = 2(8 + 5) = 26$
$LA = 26 \times 3 = 78$
$SA = 78 + 2(40) = 78 + 80 = 158 \text{ sq units}$
Example 2 — Triangular Prism
Right triangular prism: triangle base has legs 3 and 4, hypotenuse 5. Height of prism = 10.
$B = \frac{1}{2}(3)(4) = 6$
$P = 3 + 4 + 5 = 12$
$LA = 12 \times 10 = 120$
$SA = 120 + 2(6) = 132 \text{ sq units}$
Example 3 — Cylinder
Cylinder: radius 4, height 9.
$LA = 2\pi(4)(9) = 72\pi \approx 226.2$
$SA = 2\pi(4)(9) + 2\pi(16) = 72\pi + 32\pi = 104\pi \approx 326.7 \text{ sq units}$
Example 4 — Application (Label Area)
A cylindrical can has radius 3 cm and height 12 cm. Find the area of the label (lateral surface only).
$$LA = 2\pi(3)(12) = 72\pi \approx 226.2 \text{ cm}^2$$
Using Nets
A net is a two-dimensional representation of a solid that can be folded to form it.