Cube3D 立方体数理Maths · Primary school / Middle school
Why does a cylinder unfold into a rectangle?
A paper label wraps around a can without covering its circular ends. Cut straight down the label and flatten it: the curved side becomes a rectangle. The cylinder has changed shape, but the amount of paper has not changed. This is the starting point for understanding lateral area.
Find the two sides of the rectangle
One edge runs once around the circular base. Its length is the circumference, 2πr, where r is the radius. The other edge follows the height, h. Multiplying these lengths gives lateral area A = 2πrh.
For r = 2 cm and h = 3 cm, the rectangle is 4π cm long and 3 cm wide. Its area is 12π cm², approximately 37.70 cm². Keep π until the final step so that repeated rounding does not change your result.
A five-minute investigation
Before moving the controls, predict what will happen when only the radius doubles. Open the free cylinder lab, set the radius to 1 cm and height to 3 cm, and fully unfold the side. Record the lateral area.
Change the radius to 2 cm while keeping the height at 3 cm. The area changes from 6π to 12π cm²: it doubles. Now double the height as well. The area becomes four times the original because both dimensions of the rectangle have doubled.
Check your understanding
A cylinder has radius 3 cm and height 4 cm. A student says its lateral area is 12π cm². Which factor has been missed, and how would you point to it on the unfolded model?
Show the explanation
The missing factor is 2 in the circumference. The rectangle is 6π cm by 4 cm, so its area is 24π cm².
What the model assumes
The intermediate shapes illustrate unfolding. Use the fully flattened rectangle to identify the measured lengths. The model is an ideal geometric surface, with no paper thickness or stretching.