Separate space inside from material outside

1 · Learn

A right circular cylinder has the same circular cross-section all the way along its perpendicular height h. A circle of radius r has area pi × r², so stacking that constant area gives volume V = pi × r² × h. Use the radius, not the diameter, and cubic units for the space inside.

Cutting and unrolling a paper model's curved surface makes a rectangle. Its width is the circumference 2 × pi × r and its height is h, so the curved area is 2 × pi × r × h. The diagram shows this rectangle separately. This is a model: you do not need to cut a real container.

A CLOSED cylinder also has two circular ends. Total outside area is 2 × pi × r × h + 2 × pi × r². An open-top model has just one end, so add pi × r² instead. State which faces exist before adding them; the curved area alone is neither total area nor volume.

Keep pi in an exact answer. If a decimal is requested, use the calculator's pi value and round only at the end. These ideal thin-surface models ignore material thickness and seams. A radius measured in centimetres and a height in millimetres must be converted to the same unit before using the formula.

2 · Worked example

For r=3 cm and h=5 cm, one end is 9pi cm². Volume is 45pi cm³, about 141.37 cm³. Curved area is 30pi cm² and the two ends add 18pi cm², so total closed area is 48pi cm². The rectangular net's width is 6pi cm, not the diameter 6 cm.

3 · Your turn

A cylinder has radius 2 cm and height 10 cm. Find exact volume, curved surface area, and total area when closed. What changes if the top is removed?

4 · Apply your learning

On paper sketch a cylinder and its net. Label the two equal circles and a rectangle of width 2pi r and height h. Explain why knowing only the volume does not uniquely determine the radius, height or total surface area.