Nets of a cylinder and a cone

The idea

A cylinder unfolds into its two circular faces and, after a cut along its height, a rectangle from the curved surface; a cone's curved surface unrolls into a portion of a circle centred at its vertex.

A tin can

Take a tin can with a label stuck round it. Cut the label straight down the side, and open it out flat.

What shape is the label when it is flat?

• A circle
• A rectangle
• A triangle

Cut and unroll

Tap the red dashed line to cut the curved surface open. Then slide Unroll. Do it for the cylinder and for the cone.

Rectangle, and part of a circle

If the circular faces of a cylinder are unfolded, and a cut is made along the height, we get a rectangle from the curved surface. After unrolling a cone, the boundary of the net is a portion of a circle with its centre at the vertex O.

Every point on the edge of the cone's base is at the same distance from O. That is why the curved edge of the unrolled cone is part of a circle centred at O.

• Solid | Curved surface | Flat circles
• Cylinder | Rectangle | 2
• Cone | Part of a circle | 1

Notes

A cylinder gives a rectangle (cut along the height) and two circles. A cone gives a part of a circle with its centre at the vertex O, and one circle.

Check yourself

Which of these is the net of a cone?

How many flat pieces does the net of a cylinder have? Count the rectangle and the circles.

Answer: 3 pieces

1 rectangle for the curved surface + 2 circles for the top and the bottom = 3 pieces.

A paper cone is cut from its vertex O down to the edge of the base, and opened out flat. The curved edge of the flat piece is part of a circle. Where is the centre of that circle?

Meena cuts the label of a tin can straight down along its height and opens it flat. The flat label has four corners. What is true about them?

• Net A. A has a triangle, with straight sides. The curved edge of a cone unrolls to part of a circle, not to a straight line.
• Net B. B has a rectangle. That is for a cylinder. A cone comes to a point, the vertex O.
• Net C — correct. Yes! C has a part of a circle with its centre at the vertex, and the circle of the base.
• At the vertex O — correct. Yes! Every point on the edge of the base is at the same distance from O, so the edge is part of a circle centred at O.
• In the middle of the circle of the base. The base circle stays a separate piece. The curved edge of the unrolled surface is centred at the vertex.
• Nowhere: the edge is not part of a circle. It is part of a circle: all its points are at equal distances from O.
• Two of them are sharp points, like the tip of a cone. A cylinder has no sharp tip. Only a cone has the vertex O. All four corners of the flat label are alike.
• All four are right angles — correct. Yes! The height of the can stands straight up from the circle at the bottom. So the cut meets the top and the bottom edges at right angles, and the flat label is a rectangle.
• They are all different. The curved surface opens into a rectangle, so all four corners are alike. Each one is a right angle.