Volume & Surface Area of a Cylinder

Volume is the end area carried along the length. πr² is the circle at the end, and multiplying by h sweeps it through the cylinder. Every prism works this way: cross-sectional area times length, whatever the cross-section looks like.

The curved surface is a rectangle in disguise. Cut it down its length and roll it flat: one side is the height, the other is the circumference 2πr, so the area is 2πrh. That is why the formula has no square in it, while the ends do.

Radius counts twice, height only once. Doubling h doubles the volume, but doubling r quadruples it. When a question asks which change matters more, that is nearly always the point being tested.

• Try this: Set r = 3, h = 8 and note the volume. Now swap them: r = 8, h = 3. Why is the volume so different?
• Try this: Double the height and watch the volume. Then reset and double the radius instead. Which won?
• Try this: Find r and h that make the curved area and the two ends roughly equal. What is the relationship between them?