Net of a Solid — Definition, Formula & Examples
A net of a solid is a flat, two-dimensional pattern that can be cut out and folded along its edges to form a three-dimensional shape. Each face of the solid appears exactly once in the net.
A net of a polyhedron or other solid is a connected planar arrangement of the solid's faces, joined along shared edges, such that folding along those edges produces the original three-dimensional figure without overlapping any faces.
How It Works
To create a net, imagine cutting along certain edges of a 3D shape and unfolding it flat onto a table. Every face must stay connected to at least one other face, and no faces can overlap. A single solid can have multiple valid nets — a cube, for example, has 11 different nets. Nets are especially useful for calculating surface area, since you can find the area of each flat piece and add them together.
Worked Example
Problem: A rectangular prism (box) has length 4 cm, width 3 cm, and height 2 cm. Sketch its net and find the total surface area.
Identify the faces: A rectangular prism has 6 rectangular faces: two that are 4 × 3, two that are 4 × 2, and two that are 3 × 2.
Arrange the net: Lay out the six rectangles in a cross-like pattern so they share edges. For instance, place four rectangles in a row (3×2, 4×3, 3×2, 4×3) with the two 4×2 rectangles attached to the top and bottom of the second rectangle.
Calculate total surface area: Add the areas of all six faces.
Answer: The net consists of 6 rectangles, and the total surface area is 52 cm².
Why It Matters
Nets turn a 3D surface area problem into a simpler 2D area problem. They also show up in real-world design — packaging engineers use nets to create box templates that are cut from flat cardboard and folded into containers.
Common Mistakes
Mistake: Drawing a net where faces overlap when folded.
Correction: Check your net by mentally (or physically) folding it. Every face must occupy its own space on the solid with no overlapping regions.