3D Shapes & Nets Cheat Sheet

A printable reference covering prisms, pyramids, cylinders, cones, spheres, nets, faces, edges, vertices, surface area, and volume for grades 4-8.

Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

This cheat sheet covers common three-dimensional solids and how their flat nets fold into shapes. Students use it to compare prisms, pyramids, cylinders, cones, and spheres by faces, edges, vertices, surface area, and volume. It is helpful for recognizing shapes in diagrams, solving measurement problems, and checking whether a net can form a solid. The most important ideas are that surface area measures the outside covering of a solid, while volume measures the space inside it. Prisms and cylinders use the idea of a base area times height, so $V = Bh$ . Pyramids and cones have one-third the volume of a matching prism or cylinder, so $V = \frac{1}{3}Bh$ . Nets show every face of a solid laid flat, and correct nets must have the right number and arrangement of faces.

Key Facts

A prism has two congruent parallel bases, and its volume is $V = Bh$ , where $B$ is the area of one base and $h$ is the height.
A rectangular prism has volume $V = lwh$ and surface area $SA = 2lw + 2lh + 2wh$ .
A cube has volume $V = s^3$ and surface area $SA = 6s^2$ , where $s$ is the side length.
A pyramid has volume $V = \frac{1}{3}Bh$ , where $B$ is the base area and $h$ is the perpendicular height.
A cylinder has volume $V = \pi r^2h$ and surface area $SA = 2\pi r^2 + 2\pi rh$ .
A cone has volume $V = \frac{1}{3}\pi r^2h$ and surface area $SA = \pi r^2 + \pi rl$ , where $l$ is the slant height.
A sphere has volume $V = \frac{4}{3}\pi r^3$ and surface area $SA = 4\pi r^2$ .
Euler's formula for many polyhedra is $F + V = E + 2$ , where $F$ is faces, $V$ is vertices, and $E$ is edges.

Vocabulary

Net: A net is a two-dimensional pattern that can be folded along edges to form a three-dimensional solid.
Face: A face is a flat surface of a three-dimensional solid, such as one square side of a cube.
Edge: An edge is a line segment where two faces of a three-dimensional solid meet.
Vertex: A vertex is a corner point where edges of a three-dimensional solid meet.
Surface Area: Surface area is the total area of all outside faces or curved surfaces of a three-dimensional solid.
Volume: Volume is the amount of space inside a three-dimensional solid, measured in cubic units.

Common Mistakes to Avoid

Counting curved surfaces as flat faces, which is wrong because faces are usually flat polygons while cylinders and cones also have curved surfaces.
Using slant height instead of perpendicular height for volume, which is wrong because formulas like $V = \frac{1}{3}Bh$ and $V = \frac{1}{3}\pi r^2h$ require vertical height.
Forgetting units are squared for surface area and cubed for volume, which is wrong because area measures covering in square units and volume measures space in cubic units.
Assuming any group of attached polygons is a valid net, which is wrong because the faces must fold without overlapping and must meet in the correct positions.
Mixing up radius and diameter, which is wrong because circle formulas use radius and $r = \frac{d}{2}$ .

Practice Questions

1 A rectangular prism has length $8\text{ cm}$ , width $3\text{ cm}$ , and height $5\text{ cm}$ . Find its volume and surface area.
2 A cube has side length $6\text{ in}$ . Find its volume and surface area.
3 A cylinder has radius $4\text{ m}$ and height $10\text{ m}$ . Find its volume in terms of $\pi$ .
4 A net has six congruent squares arranged so that four squares form a row and one square is attached above and below the second square. Explain whether it can fold into a cube and why.

Sign in to save