A net of a cylinder is what you get when you “unfold” a 3D cylinder into a flat, 2D shape. It consists of exactly three pieces: one rectangle and two identical circles. The circles represent the top and bottom bases, and the rectangle is the curved surface laid flat. If you cut these shapes out and folded them back together, you’d reconstruct the original cylinder.
The Three Pieces of a Cylinder Net
Picture peeling the label off a tin can, then popping off the top and bottom lids. You now have three separate flat shapes sitting on a table. The two circular lids are identical, each with the same radius as the original cylinder. The label, once flattened, forms a rectangle.
In a standard diagram, the rectangle sits in the middle with one circle attached to its top edge and another to its bottom edge. But the circles can technically attach anywhere along those edges. What matters is the relationship between the rectangle’s dimensions and the circles’ size.
Why the Rectangle’s Length Equals the Circumference
This is the key detail that makes a cylinder net click. The rectangle’s length is exactly equal to the circumference of the circular base. Think about it: when you wrap that rectangle back around to form the curved surface, its edges need to meet perfectly around the circle. If the rectangle were any longer or shorter, it wouldn’t line up.
Since circumference equals 2πr (where r is the radius of the base), the rectangle in a cylinder net always has a length of 2πr. The rectangle’s other dimension, its width, equals the height of the cylinder (h). So for a cylinder with a radius of 5 cm and a height of 10 cm, the rectangle in its net would measure approximately 31.4 cm long (2 × π × 5) by 10 cm wide.
You can test this yourself with a paper towel roll. Cut straight down the side, unroll it flat, and you’ll get a rectangle. Measure its length, then measure around the circular opening. They match.
Using the Net to Calculate Surface Area
The net makes surface area intuitive because you can see every face of the cylinder laid out flat. Total surface area is just the combined area of all three pieces.
- Each circular base: πr²
- The rectangle (curved surface): 2πr × h
Add them together and you get the total surface area formula: 2πr² + 2πrh. The first part (2πr²) accounts for both circles. The second part (2πrh) is the lateral surface area, meaning just the curved “wrapper” around the cylinder without the top or bottom.
For example, a cylinder with a radius of 3 cm and a height of 7 cm has a lateral surface area of 2 × π × 3 × 7 ≈ 131.9 cm², and a total surface area of 2 × π × 9 + 131.9 ≈ 188.5 cm². The net lets you see exactly where each part of that number comes from, which is why teachers often use nets to introduce surface area concepts.
How Cylinder Nets Differ From Other Shapes
Most geometric nets are made entirely of polygons. A cube’s net has six squares, a triangular prism’s net has two triangles and three rectangles. A cylinder is unusual because it mixes curved shapes (circles) with a rectangle. This is worth keeping in mind if you’re cutting one out of paper: the circles need to be precisely sized relative to the rectangle, or the edges won’t align when you try to fold it into a cylinder.
It’s also worth noting that this standard net applies to a right circular cylinder, where the sides are perpendicular to the base. This is the common cylinder you encounter in everyday life, from cans to pipes to drinking glasses. If a cylinder is tilted (oblique), the unfolded curved surface would not form a clean rectangle, making the net more complex.
Drawing a Cylinder Net
To draw an accurate net, start with what you know: the radius (r) and height (h) of the cylinder.
First, draw a rectangle. Make its length equal to 2πr and its height equal to h. Then draw two circles, each with radius r, and attach one to the top edge and one to the bottom edge of the rectangle. Center each circle on its respective edge so they’re positioned symmetrically.
If you’re working with specific measurements, calculate 2πr first. For a cylinder with a 4 cm radius, the rectangle’s length would be 2 × 3.14159 × 4 ≈ 25.1 cm. A common mistake is using the diameter instead of the circumference for the rectangle’s length, which would give you a rectangle far too short to wrap around the base.
When folding the net into a 3D shape, the rectangle curves around to form the lateral surface, and the two circles close off the top and bottom. The fact that you can reverse this process, going from 3D to flat and back, is exactly what makes nets useful for understanding solid geometry.