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

Nets of Prisms and Pyramids

E2.5About 45 minutes

Learning goals

I can draw and recognize nets of cubes, prisms and pyramids.
I can describe the faces of a prism or a pyramid from its net.
I can explain why the total area of a net is the surface area of the object.

Words to know

net
A flat pattern that folds up to make a 3D object, with every face showing once and no overlaps.

face
A flat surface of a 3D object.

prism
A 3D object with two congruent, parallel bases joined by rectangles. It is named after its base, like a triangular prism.

pyramid
A 3D object with one base and triangles that meet at a point at the top. It is named after its base, like a square-based pyramid.

surface area
The total area of all the faces of a 3D object, in square units.

Let's start with a cereal box

Take an empty cereal box and carefully open it along its glued edges (ask a grown-up first). Lay it flat. The flat shape is a net: a pattern that folds up into the box.

A net shows every face of an object, laid out flat and joined at the edges. That makes nets very useful: to find how much cardboard covers a box, you just find the area of its net. That total area is called the surface area.

Left: a net of a cube, made of 6 congruent squares in a cross shape. Right: a net of a rectangular prism that is 3 cm long, 2 cm wide and 1 cm tall (drawn in proportion), made of 3 pairs of matching rectangles.

Faces of prisms and pyramids

A prism has 2 congruent bases and a rectangle for each side of the base. A triangular prism has 2 triangles and 3 rectangles: 5 faces.
A pyramid has 1 base and a triangle for each side of the base. A square-based pyramid has 1 square and 4 triangles: 5 faces.

So the shape of the base tells you everything: count its sides, and you know how many rectangles (prism) or triangles (pyramid) there are.

Nets of a triangular prism (3 rectangles with a triangle on each end of the middle one), a square-based pyramid (a square with a triangle on each side) and a triangular pyramid (4 equilateral triangles).

Object	Bases	Other faces	Total faces
Cube	(6 squares)	—	6
Rectangular prism	2 rectangles	4 rectangles	6
Triangular prism	2 triangles	3 rectangles	5
Pentagonal prism	2 pentagons	5 rectangles	7
Square-based pyramid	1 square	4 triangles	5
Triangular pyramid	1 triangle	3 triangles	4
Hexagonal pyramid	1 hexagon	6 triangles	7

Faces of some prisms and pyramids.

Worked example

Example 1: Read a net

A net has 2 congruent triangles and 3 rectangles. What object does it make? How many faces does it have?

Two congruent bases joined by rectangles means a prism.
The bases are triangles, so it's a triangular prism.
2 + 3 = 5 faces.

A triangular prism with 5 faces.

Three arrangements of 6 squares. A (a row of 4 with one square above the right end and one below the left end) folds into a cube. B (a 2-by-3 block) does not. C (a staircase of 2, 2 and 2) folds into a cube.

Worked example

Example 2: Net or not?

Which of the arrangements A, B and C fold into a cube?

A: fold the row of 4 into a loop of 4 sides. The square above the right end becomes the top and the square below the left end becomes the bottom. ✓
B: it has a 2-by-2 block of squares. At a cube's corner only 3 faces meet, so 4 squares around one point can't all fold up. ✗
C: imagine folding it. Each pair of squares wraps around a different edge, and all 6 faces fit with no overlap. ✓ (Check by cutting it out!)

A and C are nets of a cube. B is not.

Worked example

Example 3: Area of a net

Find the total area of the net of the rectangular prism in the first picture (3 cm by 2 cm by 1 cm).

Top and bottom: 3 × 2 = 6 cm² each, so 12 cm².
Front and back: 3 × 1 = 3 cm² each, so 6 cm².
The two ends: 2 × 1 = 2 cm² each, so 4 cm².
Total: 12 + 6 + 4 = 22 cm².

22 cm². This is the surface area of the prism, because the net is made of exactly the faces of the prism.

Just for fun

Funny but true: a cube has 11 different nets. That's 11 ways to unfold the same box, so a cube-shaped gift is never short on options!

Imagine, then test

Picturing a net folding up in your head is hard at first, and that's normal. Your brain gets better with practice. Make your guess, then test it by cutting out the net and folding it. Each test teaches you something.

Hands-on activity

Net builders

You need: grid paper, a ruler, a pencil, scissors (with a grown-up's help), tape, and an empty box from the recycling bin (like a cereal box or a toothpaste box).

With a grown-up, carefully open the box along its glued edges and flatten it. Trace around it. Ignore the little glue flaps. Which faces match?
On grid paper, draw the three arrangements A, B and C from the lesson, with squares 3 cm on each side. Cut them out and try folding each into a cube. Were your predictions right?
Try to find more cube nets. There are 11 different ones altogether (not counting flips and turns). How many can you find?
Draw a net for a triangular prism: 3 rectangles 4 cm by 6 cm in a row, with an equilateral triangle (4 cm sides) on each end of the middle rectangle. Cut and fold it.

Math talk: "How did you know a net wouldn't work before you folded it?"

Your turn

How many faces does a square-based pyramid have? What shapes are they?

How many faces does a hexagonal prism have?

How many faces does a pentagonal pyramid have?

Is a 2-by-3 block of 6 squares a net of a cube? Why or why not?

Each face of a cube is a square with 2 cm sides. What is the total area of its net?

A box is a rectangular prism. How many pairs of matching rectangles does its net have?

More practice: warm-up

What is a net?

How many faces does a cube have?

What shape are the bases of a triangular prism?

How many bases does a pyramid have?

How many triangles are in the net of a triangular pyramid?

More practice: on your own

A pyramid has a hexagon for its base. How many triangles are in its net?

A prism has 7 faces. What shape is its base? Explain.

Each face of a cube has an area of 16 cm². What is the cube's surface area?

Six squares are arranged as a row of 4, with one square above the left end and one square below the left end. Is it a net of a cube?

A net has 2 triangles and 3 rectangles. Name the object and give one real-life example.

Challenge

Six squares are arranged as a row of 5 with one square above the middle one. Can it fold into a cube? Explain.

Sketch a net for a box that is 4 cm long, 3 cm wide and 2 cm tall, and find its total area.

The net of a square-based pyramid has a square with 6 cm sides and 4 identical triangles. Each triangle has a base of 6 cm and a height of 5 cm. Find the total area of the net.

Check your understanding

Quiz

Which object has a net made of 2 triangles and 3 rectangles?

How many faces does a pentagonal prism have?

The net of a pyramid has 1 square and 4 triangles. What is it?

Which arrangement of 6 squares can never be a cube net?

What does the total area of a net tell you?

Each face of a cube has an area of 9 cm². What is the area of its net?

For parents and teachers

Big idea: The Ontario curriculum (E2.5) asks Grade 6 students to create and use nets to demonstrate the relationship between the faces of prisms and pyramids and their surface areas. The next lesson calculates surface areas.

Common difficulties:

Visualizing folding. This takes time. Cutting out and folding real nets is the best help.
Counting faces. Remind your child: prisms have 2 bases plus one rectangle per side of the base; pyramids have 1 base plus one triangle per side.
Thinking any 6 squares make a cube. Only 11 arrangements work; any arrangement with a 2-by-2 block or a row of 5 fails.

How to help: Open up empty boxes from the recycling bin together (you do any cutting) and lay them flat. Ask your child to predict which faces will match before you measure.