12.2 Surface area of prisms, cylinders, pyramids, and cones

Surface area formulas let you calculate the total area covering the outside of a 3D shape. This matters any time you need to figure out how much material wraps around a solid, whether that's paint on a wall, cardboard for a box, or metal sheeting on a roof. The formulas differ depending on the shape, so this section covers prisms, cylinders, pyramids, and cones.

Surface Area Formulas

more resources to help you study

practice questions

Formulas for prism and cylinder surfaces

The surface area of any solid is the sum of all its outer faces. For prisms and cylinders, you have two bases (top and bottom) plus the lateral area (the faces that wrap around the sides).

Prisms

The lateral area of a prism is the combined area of all the non-base faces. Since those faces wrap around the perimeter of the base, the formula is:

$L = ph$

where $p$ is the perimeter of the base and $h$ is the height of the prism.

To get total surface area, add the two bases:

$SA = 2B + ph$

where $B$ is the area of one base.

For example, a rectangular prism with a 3 cm × 5 cm base and a height of 10 cm has $p = 16$ cm and $B = 15$ cm². So $SA = 2(15) + 16(10) = 30 + 160 = 190$ cm².

Cylinders

A cylinder works the same way, except the base is a circle and the lateral surface "unrolls" into a rectangle.

• Lateral area: $L = 2\pi rh$
• Total surface area: $SA = 2\pi rh + 2\pi r^2$

The $2\pi r^2$ term accounts for the two circular bases. Think of it this way: if you peel the label off a can, that rectangle has a width of $h$ and a length equal to the circumference $2\pi r$.

Surface area of pyramids and cones

Pyramids and cones have only one base, and their lateral faces slant inward to a point (the apex). Because of that slant, these formulas use slant height ($s$) instead of the vertical height.

Pyramids

• Lateral area: $L = \frac{1}{2}ps$, where $p$ is the perimeter of the base and $s$ is the slant height
• Total surface area: $SA = \frac{1}{2}ps + B$, where $B$ is the area of the base

This formula assumes a regular pyramid (the base is a regular polygon and the apex is centered directly above it), so all lateral faces are congruent triangles.

Cones

• Lateral area: $L = \pi rs$, where $r$ is the radius of the base and $s$ is the slant height
• Total surface area: $SA = \pi rs + \pi r^2$

The $\pi r^2$ term is the single circular base. If a cone has $r = 4$ cm and $s = 9$ cm, then $SA = \pi(4)(9) + \pi(4)^2 = 36\pi + 16\pi = 52\pi \approx 163.4$ cm².

Problem Solving and Applications

Real-world surface area applications

When you hit a word problem, follow these steps:

1. Identify the solid. Is it a prism, cylinder, pyramid, or cone? A soup can is a cylinder; a tent might be a cone or pyramid.
2. List the dimensions you're given (base lengths, radius, height, slant height). If slant height is missing for a pyramid or cone, calculate it using the Pythagorean theorem.
3. Decide if you need total or lateral surface area. Painting the outside of a closed box? Total SA. Wrapping just the sides of a cylinder with no lid? Lateral area plus one base.
4. Plug values into the correct formula and compute.
5. Interpret your answer in context, with correct units (always squared units for area).

Effects of dimension changes on surface area

Understanding how scaling dimensions affects surface area helps you predict results without recalculating everything from scratch.

Prisms and cylinders:

• Doubling the height doubles the lateral area (since $L = ph$ or $L = 2\pi rh$), but the base areas stay the same. Total SA increases, but it does not double overall.
• Doubling all base dimensions (length and width, or radius) quadruples the base area. The perimeter (or circumference) also doubles, so the lateral area doubles as well.

Pyramids and cones:

• Doubling the vertical height changes the slant height (since $s = \sqrt{r^2 + h^2}$ for a cone, for instance), but it does not simply double $s$. You need to recalculate.
• Doubling all base dimensions quadruples the base area and doubles the perimeter (or circumference), which doubles the lateral area.

General scaling rule: If you multiply every dimension of a solid by a factor of $k$, the surface area is multiplied by $k^2$. So tripling all dimensions makes the surface area $9$ times larger. This is a powerful shortcut for comparison problems.