Solve problems involving surface area of solids (e.g., prisms, pyramids, cylinders). | Area and volume | SAT Math

Surface area is the sum of all outside faces, with special attention to slant height and whether the base is included.

Core Idea

Surface area is the total area of every face of a 3D solid. Think of it as the amount of wrapping paper you'd need to cover the object completely.

Understanding

For a rectangular prism (box), add the areas of all six faces — three pairs of identical rectangles. For a cylinder, the surface area is $2 𝜋 𝑟 2 + 2 𝜋 𝑟 ℎ$ : two circular caps plus a rectangular wrap whose width is the circumference $2 𝜋 𝑟$ .

Cones and pyramids have a lateral (slant) surface plus a base. The SAT reference sheet lists these, so focus on reading the right values off the problem — especially distinguishing the slant height from the vertical height.

If the problem asks for only the lateral surface area, leave out the base(s).

Step by Step

Identify the solid: box, cylinder, cone, pyramid, or sphere.
Decide whether the problem wants total surface area or just the lateral surface.
List all dimensions — watch for radius vs. diameter and slant height vs. vertical height.
Plug into the formula and simplify.

Misconceptions

Using vertical height instead of slant height in cone or pyramid lateral area formulas.
Forgetting to include both circular bases of a cylinder when the problem asks for total surface area.
Mixing up $2 𝜋 𝑟 ℎ$ (lateral area of a cylinder) with $𝜋 𝑟 2 ℎ$ (volume of a cylinder).

Question

Worked Example

A closed cylindrical container has a radius of 3 inches and a height of 10 inches. What is the total surface area of the container, in square inches?

Select an answer to see the explanation