Lesson 55: Nets of Prisms, Cylinders, Pyramids, and Cones

New Concept

If we think of the surface of a solid as a hollow cardboard shell, then cutting open and spreading out the cardboard creates a net of the solid.

What’s next

Soon, you'll practice sketching nets for different solids and see how they connect directly to their surface area formulas in worked examples.

Property

If we think of the surface of a solid as a hollow cardboard shell, then cutting open and spreading out the cardboard creates a net of the solid.

Examples

• The net for a cube is a cross shape made of six identical squares, which fold up to form the six faces.
• A net for a pyramid with a square base is a central square with four triangles attached to its sides.
• Unfolding a standard cardboard box creates a net showing all six of its rectangular faces.

Explanation

Imagine carefully cutting open a cereal box and laying it flat—that's a net! It’s the 2D blueprint showing all the faces of a 3D shape connected together. This flat pattern helps us easily see, understand, and calculate the total surface area of the solid before it is folded up into its cool 3D form.

Property

The net of a cone has two parts: its circular base and a sector of a circle that forms the lateral surface. The surface area formula is $S = \pi r l + \pi r^2$.

Examples

• A cone's net consists of a circle and a larger circular sector, which looks like a slice of pizza.
• The term $\pi r^2$ in the formula $S = \pi r l + \pi r^2$ represents the area of the flat circular base.
• The term $\pi r l$ calculates the area of the lateral surface, the part that wraps around to make the cone's pointy body.

Explanation

Think of an ice cream cone! Its net is like a party hat shape (a sector of a circle) plus a flat circle for the lid. The curvy part wraps around to make the cone's body, and the circle is its base. The formula simply adds the area of the circular base ($\pi r^2$) and the lateral surface ($\pi r l$).

Property

A cylinder has two circular bases and a lateral surface that unwraps to form a rectangle. The surface area is given by the formula $S = 2\pi rh + 2\pi r^2$.

Examples

• A cylinder's net is a rectangle with two identical circles attached to its shorter, opposite sides.
• In the formula $S = 2\pi rh + 2\pi r^2$, the $2\pi rh$ part calculates the area of the rectangular label.
• The $2\pi r^2$ part of the formula represents the combined area of the two circular lids on the top and bottom.

Explanation

A can of soup is a perfect cylinder. If you were to peel off its label, the label would become a flat rectangle. The top and bottom lids of the can are just two circles. So, a cylinder's net is simply one rectangle with two circles attached, one at the top and one at the bottom.