These problems ask you to find the length of a string wrapped around a cylinder or prism. The key is to make a net (an unfolded figure) and think of the string as a “straight line.”
How to Solve String Problems
- Cylinder: When the lateral surface is unfolded, it becomes a rectangle. The string becomes a diagonal line of that rectangle.
- Length of the string = length of the diagonal line in the net (use the properties of right triangles, or the Pythagorean theorem)
- Rectangular prism: Unfold the faces so that the string becomes a “straight line.”
Example: A string is wrapped once from the bottom to the top of a cylinder whose base has a circumference of 12 cm and whose height is 5 cm. What is the shortest possible length of the string?
Solution:
① When the lateral surface of the cylinder is unfolded, it becomes a rectangle 12 cm wide × 5 cm high
② The string becomes the diagonal line of that rectangle
③ By the Pythagorean theorem, √(12² + 5²) = √(144 + 25) = √169 = 13 cm
Solution:
① When the lateral surface of the cylinder is unfolded, it becomes a rectangle 12 cm wide × 5 cm high
② The string becomes the diagonal line of that rectangle
③ By the Pythagorean theorem, √(12² + 5²) = √(144 + 25) = √169 = 13 cm
When you are asked for the “shortest length,” remember to make a net and draw a straight line. For a cylinder, always use the Pythagorean theorem (the property of right triangles).