Grade by Grade
E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

Building 3D Objects from Top, Front and Side Views

E1.2 A1About 45 minutes
Learning goals
  • I can build a structure of cubes when I am given its top, front and side views.
  • I can write a top view with numbers to record how tall each stack is.
  • I can find the fewest and the most cubes that fit a set of views, and name objects from their views.

Words to know

view
A flat drawing of an object as you see it looking straight at one side: from the top, the front or the side.
top view with numbers
A top view where each square shows how many cubes are stacked on that spot. It is a complete building plan. (Some people call it a mat plan.)
right side view
The view from the right side. The front of the object is on the left of this drawing.
three-dimensional (3D) object
An object with length, width and height, like a box, a can or a building.
prism
A 3D object with two matching, parallel bases joined by rectangle faces.

Remember from Grade 5

In Grade 5 you drew the top, front and right side views of objects and of structures made of cubes. You found out that one view can fool you, so designers use several views. This year you will go the other way: you get the views, and you build the object.

Let's start with a builder

A construction crew in your town gets drawings of a new library. Nobody gives them a 3D model. They have to read the top, front and side views and imagine the building. Today you are the builder!

Remember what each view tells you:

  • The top view shows which spots on the table have cubes. It doesn't show how tall the stacks are.
  • The front view shows the tallest stack in each column, from left to right.
  • The right side view shows the tallest stack in each row, from front (on the left) to back (on the right).
the structure2111topfrontright side
A structure of 5 cubes and its views. The top view with numbers shows the stacks: 2, 1, 1 in the front row and 1 behind the right stack. The front view shows columns 2, 1 and 1 cubes tall. The right side view shows the front row (left) 2 cubes tall and the back row (right) 1 cube tall.

How to build from views

  1. Top view first. Put 1 cube on every square of the top view.
  2. Front view. Look at each column from left to right. The front view tells you how tall the tallest stack in that column must be.
  3. Side view. Look at each row from front to back. The side view tells you how tall the tallest stack in that row must be.
  4. Add cubes until every stack matches both views. Then check all three views again.
  5. Record your building as a top view with numbers.
Worked example

Example 1: Build from three views

The top view is an L: 3 squares in the front row and 1 behind the right square. The front view has columns of 2, 1 and 1. The right side view has 2 (front) and 1 (back). Build it.

  1. Top view: put 1 cube on each of the 4 squares.
  2. Front view, left column: it must be 2 tall. The left column has only one square (front left), so that stack gets a second cube.
  3. Front view, middle and right columns: only 1 tall. So the front-middle, front-right and back-right stacks stay at 1.
  4. Side view: the front row must be 2 tall (it is, because of the front-left stack) and the back row 1 tall (it is).
  5. Check: front 2, 1, 1 ✓. Side 2, 1 ✓. Top ✓.

Top view with numbers: front row 2, 1, 1, back row 1 (on the right). That's 5 cubes, like the picture above.

topfrontright side2112fewest cubes: 6
Three views that are each a 2-by-2 square. On the right is a top view with numbers for one structure that fits them using the fewest cubes: stacks of 2 at the front left and back right, and single cubes in the other two spots.
Worked example

Example 2: Fewest and most cubes

A structure's top view, front view and right side view are each a 2-by-2 square of squares. What are the fewest and the most cubes it could have?

  1. The top view says there are 4 stacks, in a 2-by-2 square.
  2. Most: every stack can be 2 tall. 4 × 2 = 8 cubes (a big cube).
  3. Fewest: each column needs at least one stack of 2, and each row needs at least one stack of 2. Put a stack of 2 at the front left and the back right. That covers both columns and both rows.
  4. The other 2 spots need only 1 cube each. 2 + 2 + 1 + 1 = 6 cubes.

Fewest 6 cubes, most 8 cubes. The views alone don't always tell you exactly how many cubes there are, because cubes can hide behind each other.

Objects that aren't made of cubes

Views work for any object. Look at each view and ask: "What 3D shape would look like this from here?"

topfrontright side
Three views of one object. Top: a rectangle with a line down the middle from front to back. Front: a house shape (a pentagon). Right side: a tall rectangle with a line across it.
Worked example

Example 3: Name the object

Use the three views in the picture above. What object could it be?

  1. The front view looks like a house: a rectangle with a triangle on top.
  2. The side view is a rectangle, with a line where the walls end and the roof begins. So the triangle shape goes straight back like a roof: that's a triangular prism lying on its side.
  3. The top view shows the two slopes of the roof, meeting at the ridge line down the middle.

A rectangular prism with a triangular prism on top, like a house or a shed with a peaked roof.

Just for fun

What did the cube structure say to the builder who only looked at its top view?

"You don't know the half of me!"

Check every view

After you build, walk around your structure. Crouch down so your eyes are level with the front, then with the right side, then look straight down from above. If one view is wrong, fix one stack at a time. Builders call this checking your work, and it's part of being a good problem-solver.

Hands-on activity

Builder's challenge

You need: 10 to 20 building blocks, interlocking bricks or sugar cubes (all the same size), grid paper and a pencil. Keep small pieces away from little children.

  1. Your grown-up secretly builds a structure with 5 to 8 blocks, then hides it behind a cereal box.
  2. They draw its top, front and right side views on grid paper.
  3. You build a structure that matches all three views. Write your building as a top view with numbers.
  4. Compare with the hidden structure. Is yours the same? If not, does yours still match all three views? (Sometimes more than one building fits!)
  5. Switch jobs: you build and draw, and your grown-up builds.

Math talk: "Which view did you use first? Which view helped you most?"

Your turn

1

A top view with numbers has a front row of 1, 2, 1 and a 1 behind the middle stack. Describe the front view and the right side view. How many cubes are there?

2

The top view is 3 squares in a row. The front view has columns of 1, 3 and 1. How many cubes are there, and how are they arranged?

3

In Example 2, all three views are 2-by-2 squares. What is the most cubes the structure could have?

4

An object's top view is a square with lines from each corner to the centre. Its front and side views are triangles. What is it?

5

The top view is an L: 3 squares in the front row and 1 square behind the left one. The front view has columns of 3, 1 and 1. The right side view has 1 (front) and 3 (back). Write the top view with numbers.

6

Why do builders need more than one view?

More practice: warm-up

1

In a top view with numbers, what does each number tell you?

2

A top view with numbers shows 2 and 3 side by side. How many cubes are there?

3

A front view has columns 1 and 2 tall. How tall is the tallest stack in the right column?

4

In a right side view, is the front of the structure on the left or the right of the drawing?

5

Every square in a 2-by-3 top view with numbers says 1. How many cubes are there?

More practice: on your own

1

The top view is a U: 3 squares in the front row, plus 1 behind the left square and 1 behind the right square. The front view has columns of 2, 1, 2. The right side view has 2 (front) and 1 (back). Write the top view with numbers.

2

The top view has 2 squares in the front row (left and middle) and 2 squares in the back row (middle and right). The front view has columns 1, 3, 1. The right side view has 2 (front) and 3 (back). Write the top view with numbers.

3

An object's top view is a rectangle with a line down the middle from front to back. Its front view is a triangle. Its side view is a rectangle. What could it be?

4

A top view with numbers has a front row of 3, 1 and a back row of 2 (left only). Describe the front view and the right side view.

5

Explain how a front view can "hide" a cube.

Challenge

1

A structure's top view is a staircase: 3 squares in the front row, 2 in the middle row (left and middle), and 1 in the back row (left). The front view has columns 3, 2, 1 and the right side view has rows 3, 2, 1 (front to back). What are the fewest and the most cubes it could have?

2

The top view is a 2-by-2 square. The front view has columns 1 (left) and 3 (right). The right side view has 1 (front) and 3 (back). Write the top view with numbers. Is there only one answer?

3

An object's top view is a square with a circle inside it. Its front view and side view are both a wide rectangle with a narrower rectangle standing on top in the middle. Describe the object.

Check your understanding

Quiz

1.

In a top view with numbers, what does a 3 on a square mean?

2.

A top view with numbers has a front row of 1, 2 and a back row of 3, 1. How many cubes are there?

3.

In a right side view, where is the front of the structure?

4.

An object's top view is a circle and its front and side views are rectangles. What could it be?

5.

All three views of a structure are 2-by-2 squares. What is the most cubes it could have?

6.

Which view shows the tallest stack in each column, from left to right?

For parents and teachers

Big idea: The Ontario curriculum (E1.2) asks Grade 6 students to construct three-dimensional objects when given their top, front and side views. In Grade 5 they drew views; now they work backwards, which takes strong spatial reasoning.

Common difficulties:

  • Left and right in the side view. In a right side view, the front of the structure is on the left. Have your child physically walk to the right side and look.
  • Thinking views tell everything. Cubes can hide behind each other, so several structures may fit the same views. That's why questions ask for the fewest or most cubes.
  • Building without checking. Encourage checking all three views at the end.

How to help: Blocks, interlocking bricks or sugar cubes are perfect. Take turns building secret structures and drawing views for each other. Keep small pieces away from younger children.