When the curved lateral surface of a cylinder is cut along one edge and unrolled flat, what shape does it form?
AA circle
BA rectangle
CA triangle
DA trapezoid
The lateral surface of a cylinder — the 'side' of the can — unrolls into a flat rectangle. Its height equals the height of the cylinder (h), and its width equals the distance around the circular base, which is the circumference: 2πr. This is the key insight that makes the surface area formula understandable rather than memorized: the lateral area is just the area of that rectangle, width × height = 2πr × h = 2πrh. Visualizing or physically demonstrating this unrolling is the best way to internalize why the formula works.
Question 2 Multiple Choice
A cylinder has radius 3 and height 5. What is its total surface area?
Total surface area = 2πr² + 2πrh. With r = 3 and h = 5: two bases = 2π(9) = 18π; lateral surface = 2π(3)(5) = 30π; total = 48π. Option A (30π) is the common error of computing only the lateral surface and forgetting both circular bases — thinking of the cylinder as 'just the side.' Option C (39π) is the error of including only one base instead of two. The formula must include all three faces: top, bottom, and the unrolled side.
Question 3 True / False
The width of the rectangle formed by unrolling a cylinder's lateral surface equals the circumference of the circular base.
TTrue
FFalse
Answer: True
When you cut the lateral surface of a cylinder along a vertical line and unroll it, you travel exactly once around the circular base — a distance equal to the circumference, 2πr. This becomes the width of the resulting rectangle, while the height of the cylinder becomes the rectangle's height. Area = width × height = 2πr × h = 2πrh. Understanding why the width equals the circumference (not the diameter, not the radius) is the key to not just memorizing the formula but understanding it.
Question 4 True / False
The lateral surface area of a cylinder with radius r and height h is πrh.
TTrue
FFalse
Answer: False
The correct formula for lateral surface area is 2πrh, not πrh. The factor of 2 comes from the circumference of the circle, which is 2πr — this is the width of the rectangle the lateral surface unrolls into. The area is then 2πr × h = 2πrh. A common error is using πr (the radius times π without the factor of 2) instead of 2πr (the circumference). This typically happens when students confuse the radius with the circumference, or when they use πd and substitute r instead of d.
Question 5 Short Answer
Explain in your own words why the formula for the lateral surface area of a cylinder is 2πrh. Where does each part of the formula come from?
Think about your answer, then reveal below.
Model answer: If you cut the curved side of a cylinder along a straight vertical line and unroll it flat, you get a rectangle. The height of that rectangle is the same as the height h of the cylinder. The width of the rectangle is the distance all the way around the circular base — the circumference — which equals 2πr. The area of this rectangle is width × height = 2πr × h = 2πrh. So the formula comes directly from the fact that the lateral surface is a rectangle whose width equals the base's circumference.
Students who understand the unrolling insight can reconstruct the formula even if they forget it on a test. Those who only memorize '2πrh' are helpless when asked to derive or explain it. The formula also explains why both r and h matter equally in the lateral term: the width of the rectangle depends on r (through the circumference), and the height of the rectangle is h.