# Views of cube buildings · Grade 12 · 3 units

> Views of cube buildings, Grade 12 · 3 units: Counting the cubes and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Views of cube buildings (Grade 12 · 3 units) you practise 4 skills: Counting the cubes · The views · From the views to the building · The least and the most.

A building of cubes is described by a numeric diagram: look at the building from above, and in each square write how many cubes are in the column on it. An empty square has no cubes. The number of cubes in the building is the sum of the numbers in the diagram.

A view of the building is what is seen from one side. From the front, each column of squares in the diagram shows as one column, as tall as the largest number in it. From the right, each row shows as one column, as tall as the largest number in the row, with the front row on the left. From above, the view shows which squares are taken. The back view is the front view with left and right swapped.

Several different buildings can look the same from several sides. From the front view and the right view, find the least and the greatest number of cubes the building can have: in the largest building every square is as tall as both views allow, and in the smallest one column serves both views wherever it can.

**Tip:** Before answering, mark which side of the diagram is the front and which is the right, and only then read the view.

## What you learn

### Counting the cubes

- Example: (The numeric diagram of the building, row by row from the front: 2, 1; 0, 3; 2, 1) The diagram shows a building of cubes from above. The number in each square is the number of cubes in the column on it, and the front of the building is at the bottom of the diagram. How many cubes is the building made of?
- Answer: 9
- Solution: Add the numbers in the diagram: 2 + 1 + 3 + 2 + 1 = 9 .

### The views

- Example: (The numeric diagram of the building, row by row from the front: 0, 3, 2; 1, 0, 3) The diagram shows a building of cubes from above. The number in each square is the number of cubes in the column on it, and the front of the building is at the bottom of the diagram. The building is seen from the front. How many cubes tall is the second column from the left in this view?
- Answer: 3
- Solution: This column is the second column of squares from the left in the diagram. Its only number is 3, and from the front the tallest of its columns shows: 3 cubes.

### From the views to the building

- Example: (The plan view, row by row from the front and left to right, 1 for a square taken and 0 for an empty one: 1, 1; 0, 1; 1, 1) (The front view: columns 2 and 3 high) Here are the plan view and the front view of a building of cubes, and three numeric diagrams. Picture A: (Picture A: a numeric diagram, row by row from the front: 2, 1; 0, 3; 2, 1) Picture B: (Picture B: a numeric diagram, row by row from the front: 2, 0; 0, 3; 2, 1) Picture C: (Picture C: a numeric diagram, row by row from the front: 2, 0; 1, 3; 2, 1) Which of the pictures fits both views?
- Options: Picture A · Picture B · Picture C
- Answer: Picture A
- Solution: In the given front view the columns are 2, 3 . In Picture B the plan view is different. In Picture C the plan view is different. Only Picture A fits every view.

### The least and the most

- Example: (The front view: columns 2 and 3 high) Here is the front view of a building of cubes. What is the least number of cubes the building can have?
- Answer: 5
- Solution: Each column of the view needs at least one column of cubes of its height, and there may be only one. So the least number is 2 + 3 = 5 .

## Not on the site yet

- Drawing a building in perspective
- Views of solids not built from cubes
- Building a model from cubes

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 3 units (grades 10 and 11).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 12 · 3 units topics](https://kixplo.com/il/en/math/grade-12-3u/)