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E1.2 · Build three-dimensional objects from top, front, and side views
Learn to build three-dimensional objects from top, front, and side views through clear examples and targeted practice.
Ontario Grade 6 Mathematics
Spatial Sense
Use top, front, and side views to make a three-dimensional object
A three-dimensional object has height, width, and depth. A view is a drawing that shows an object from one direction. In this lesson, you will use top, front, and side views to build objects from unit cubes. A unit cube is one small cube used as a single building piece. Before you begin, remember that a stack of three cubes is three cubes high, while a row of three cubes is three cubes long.
What you will learn
- Identify what a top, front, and side view shows.
- Use the squares in the views to build stacks of unit cubes.
- Check whether a three-dimensional build matches all three views.
What each view tells you
Imagine looking at a small object made from cubes. The top view shows which places are covered when you look down. It shows the object's footprint, or the places where cubes touch the floor.
The front view shows the object as you look straight at its front. It tells you how tall the stacks are from left to right. The side view shows the object from the side. It tells you how tall the stacks are from front to back.
A square in a view stands for a visible square face. In a height view, a stack that is three cubes high appears three squares high. The drawings do not show every hidden cube. You must use all three views to work out how the cubes fit together.
- Top view: look down to see the footprint.
- Front view: look from the front to compare heights across the width.
- Side view: look from the side to compare heights across the depth.
A careful way to build
First, use the top view to mark each place where a stack must stand. A place with no square has no stack. A place with a square needs at least one cube.
Next, use the front view to check the tallest stack in each left-to-right position. Use the side view to check the tallest stack in each front-to-back position. When a view shows a height of three, at least one stack in that position must reach three cubes.
Add cubes to make a build that fits all the views. Then look at your build from above, the front, and the side. Compare each view with the drawing. If they do not match, change the stacks.
Views do not always give only one possible build. For example, a front view may tell you the tallest stack in a column but not the heights of every stack behind it. If more than one build fits, each is acceptable unless the task gives more information.
- Start with the footprint, then set stack heights.
- Check maximum heights from both directions.
- A correct build must match every given view.
Guided practice and checking
Read the views in order before placing cubes. Keep track of which positions are left, right, front, and back. This helps you avoid turning the object around by mistake.
After building, count the cubes in each stack and compare the heights with the views. Do not assume that a top-view square means a stack is only one cube high. It shows a place occupied by the object, not the full height.
For a harder puzzle, a view may leave some stack heights unknown. Use the other views to narrow the choices. Pick heights that satisfy all the views, and check the completed object from each direction.
- Keep the object's orientation the same in every view.
- A top-view square marks an occupied position, not a height.
- Use all three views to check a build.
Worked example
One row of three stacks
Build an object from unit cubes. The top view has three squares in one straight row. The front view shows heights of 1, 3, and 2 cubes from left to right. The side view shows one position with a height of 3 cubes.
Schematic solid model: three adjacent stacks with heights 1, 3, and 2 cubes.
- Mark the footprintThe top view has three squares in a row, so place one stack in each of the three positions. There are no other positions to fill.
- Set the stack heightsThe front view gives each stack's height because there is only one row from front to back. Make the left stack one cube high, the middle stack three cubes high, and the right stack two cubes high.
- Check the sideFrom the side, the tallest stack is three cubes high. That matches the side view, so this build fits all three views.
Answer: Build three stacks in a row with heights 1, 3, and 2 cubes from left to right.
Check: The top view has three occupied positions, the front heights are 1, 3, and 2, and the side view's greatest height is 3.
Worked example
A two-by-two footprint
Build an object with a top view that has four squares arranged in two rows and two columns. The front view has heights 2 and 1 from left to right. The side view has heights 2 at the front and 1 at the back. Find one build that fits.
Schematic solid model: a two-by-two footprint with front-row heights 2 and 1 and back-row heights 1 and 1.
- Place four stacksThe top view has four squares, so make four stack positions: two across and two from front to back. Start each position with one cube.
- Choose stack heightsPut two cubes in the front-left position and one in the front-right position. Put one cube in each back position. The front row now reaches heights 2 and 1, and the back row has heights 1 and 1.
- Compare the viewsLooking from the front, the tallest stacks in the two columns are 2 and 1. Looking from the side, the front row reaches 2 and the back row reaches 1. Both views match. Other builds may also fit these views.
Answer: One valid build has front-row heights 2 and 1, and back-row heights 1 and 1.
Check: There is a stack in all four top-view positions. The front column heights are 2 and 1, and the front-to-back heights are 2 and 1.
Worked example
Use both views to set the tallest stacks
The top view has six squares in a two-row, three-column rectangle. The front view gives column heights 2, 3, and 1 from left to right. The side view gives a front-row height of 3 and a back-row height of 2. Build one object that matches.
Schematic solid model: front-row heights 1, 3, 1 and back-row heights 2, 1, 1.
- Fill the footprintThe top view shows two rows of three positions. Put at least one cube in each of the six positions.
- Make the front rowUse front-row heights 1, 3, and 1 from left to right. This row reaches the required column heights of 1, 3, and 1 so far. The first column still needs a stack of height 2, so add that height in the back row.
- Complete and check the back rowUse back-row heights 2, 1, and 1. The tallest stacks in the three columns are now 2, 3, and 1. The front row reaches 3, and the back row reaches 2. The build matches both height views and the six-position top view.
Answer: One valid choice is front-row heights 1, 3, and 1, with back-row heights 2, 1, and 1.
Check: The column maximum heights are 2, 3, and 1. The row maximum heights are 3 at the front and 2 at the back.
Common mistakes and how to avoid them
Treating each top-view square as a stack exactly one cube high.
Correction: A top-view square shows an occupied position. Use the front and side views to decide how tall its stack may need to be.
Matching the front view but not checking the side view.
Correction: Look at the finished build from all three directions. A build is correct only when every given view matches.
Mixing up the front and back rows.
Correction: Keep the same front of the object in mind while reading the top, front, and side views.
Lesson summary
- The top view shows the object's footprint.
- The front and side views show stack heights from different directions.
- Build stacks in the positions shown from above, then compare the heights in both other views.
- More than one object can sometimes match the same three views.
Check your understanding
Question 1
A top view has two squares in a row. The front view shows heights 1 and 2. What stack heights should you build from left to right?
- 1 and 2
- 2 and 2
- 1 and 1
- correctIndex: 0,
Show answer and explanation
1 and 2
There are two positions, and the front view gives their heights as 1 and 2 from left to right.
Question 2
A top-view square appears above one position. What does that tell you?
- The position is occupied by the object.
- The stack is exactly one cube high.
- The position must be empty.
- correctIndex: 0,
Show answer and explanation
The position is occupied by the object.
A top-view square marks an occupied position. It does not give the stack's full height.
Question 3
A build matches the top and front views but not the side view. Is it a correct build?
- Yes, because two views match.
- No, because all three views must match.
- Yes, if it uses unit cubes.
- correctIndex: 1,
Show answer and explanation
No, because all three views must match.
A correct build must match every view that is given.
Key terms
- Three-dimensional
- Having height, width, and depth.
- View
- A drawing of an object as seen from one direction.
- Footprint
- The positions covered by an object when viewed from above.
- Unit cube
- One small cube used as a single building piece.
Continue through Grade 6 Math
View the complete Grade 6 Math Ontario Grade 6 Mathematics curriculum and lessons
- E1.1 · Classify quadrilaterals using sides, diagonals, and symmetry
- E1.3 · Plot four-quadrant coordinates and describe translations
- E1.4 · Combine translations, reflections, and rotations on a grid
- E2.1 · Measure and convert metric length, area, mass, and capacity
- E2.2 · Measure and construct angles to 360 degrees
- E2.3 · Solve with complementary, supplementary, opposite, and interior angles
About this lesson and its review
Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation E1.2. It is a study resource, not an official curriculum publication.
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