GEOMETRY

Views Of 3d Shapes

Learn the reliable method for views of 3d shapes, then apply it to 11+ exam-style questions.

1 · WHAT YOU NEED TO KNOW

The idea

Plan, front and side views show a solid from one direction at a time.

Picture itBuild the solid from unit cubes
2 · THE METHOD

Follow these steps

  1. Choose the viewing direction.
  2. Project occupied positions onto that view.
  3. Use visible heights or layers without counting hidden cubes twice.
3 · WORKED EXAMPLES

See the method in action

Example 1

The plan view uses a 3 × 3 floor grid. How many positions are occupied, and how many are empty?

View from above required
  1. First count one top square at each occupied floor position: 6.
  2. Then subtract from the 3 × 3 grid: 9 − 6 = 3 empty positions.
Answer: 6 occupied; 3 empty
Example 2

The plan view uses a 3 × 3 floor grid. How many positions are occupied, and how many are empty?

View from above required
  1. First count one top square at each occupied floor position: 3.
  2. Then subtract from the 3 × 3 grid: 9 − 3 = 6 empty positions.
Answer: 3 occupied; 6 empty
4 · ELEVENSPRINT TIP

A useful shortcut

Use visible heights or layers without counting hidden cubes twice.

5 · COMMON MISTAKES

Watch out for these

6 · TRY IT YOURSELF

Three fresh questions

Ready to check the method? You will get three fresh, validated questions.