A net is a flat pattern that can be folded to make a three-dimensional solid. Nets are useful because they connect the shapes you see on paper to the objects you hold in space. They help you identify faces, edges, and bases, and they make surface area easier to calculate.
Cubes, prisms, pyramids, cylinders, and cones can all be studied by unfolding their surfaces into simpler 2D shapes.
When a solid unfolds into a net, each face keeps its true size and shape. To find surface area, calculate the area of every part of the net and add the results. Prisms and pyramids unfold into polygons, while cylinders and cones include curved surfaces that become rectangles or sectors.
A correct net must have the right faces connected in a way that folds without overlap.
Key Facts
- Surface area = sum of the areas of all faces in the net.
- Cube surface area: SA = 6s^2, where s is the side length.
- Rectangular prism surface area: SA = 2lw + 2lh + 2wh.
- Triangular prism surface area: SA = 2B + Ph, where B is the area of the triangular base, P is the base perimeter, and h is the prism length.
- Cylinder surface area: SA = 2πr^2 + 2πrh, where 2πrh is the rectangle from the curved surface.
- Cone surface area: SA = πr^2 + πrl, where l is the slant height.
Vocabulary
- Net
- A net is a two-dimensional pattern that can be folded to form a three-dimensional solid.
- Face
- A face is a flat surface of a three-dimensional shape.
- Edge
- An edge is a line segment where two faces of a solid meet.
- Surface Area
- Surface area is the total area of all outside surfaces of a three-dimensional object.
- Slant Height
- Slant height is the distance along the side of a cone or pyramid from the base to the top point.
Common Mistakes to Avoid
- Counting a missing or extra face in a net is wrong because the net must match every surface of the solid exactly once.
- Using volume formulas for surface area is wrong because surface area measures square units on the outside, not cubic units inside.
- Forgetting the two bases of a prism or cylinder is wrong because both ends are part of the outside surface unless the problem says they are open.
- Treating a cone's slant height as its vertical height is wrong because the curved surface area uses the side length l, not the straight height through the center.
Practice Questions
- 1 A cube has side length 5 cm. Draw or imagine its net and calculate its total surface area.
- 2 A rectangular prism has length 8 m, width 3 m, and height 4 m. Find the surface area using the areas of its six faces.
- 3 A student draws a cube net with six squares in a straight row. Explain why this is not a valid cube net even though it has the correct number of faces.