Surface Area

Surface Area: Wrapping the Outside

Surface area is the total area of every face on the outside of a solid (3D) shape. A helpful mental trick is imagining you could unfold the solid flat into a 2D pattern called a net — then surface area is just the total area of that net.

Rectangular Prisms

Picture a rectangular box (like a shoebox) with length l, width w, and height h. Unfolded, it has six rectangular faces that pair up into three matching pairs: a top and bottom (each l\times w), a front and back (each l \times h), and two sides (each w \times h). The total surface area is:

Cubes

A cube is just a rectangular prism where all edges have the same length s. All six faces are identical squares of area s^2, so:

Cylinders

Picture a can of soup: a cylinder has two circular faces (top and bottom, each with radius r) connected by a curved side. If you cut the curved side vertically and unroll it flat, it becomes a rectangle — its height equals the cylinder's height h, and its width equals the circumference of the circular base, 2\pi r.

Adding the two circles (each \pi r^2) to the unrolled rectangle (2\pi r \times h) gives the total surface area:

Example

Find the surface area of a cylinder with radius 3 and height 10.

1. Area of the two circular bases: 2\pi r^2 = 2\pi(3)^2 = 18\pi
2. Area of the unrolled curved side: 2\pi r h = 2\pi(3)(10) = 60\pi
3. Add them together: SA = 18\pi + 60\pi = 78\pi

The cylinder's total surface area is 78\pi square units.

Key Takeaways

• Surface area = total area of all outer faces; imagining the unfolded net helps organize the pieces.
• Rectangular prism: SA = 2lw+2lh+2wh.
• Cube: SA = 6s^2.
• Cylinder: SA = 2\pi r^2 + 2\pi r h (two circles plus an unrolled rectangle).