Minds On

Inferring and observing shapes on sides

You will be working with three-dimensional objects.

For each of the following objects, count how many rectangles, triangles, squares, or trapezoids you infer and observe on the sides of the three-dimensional objects. Consider all the sides: top, bottom, right, left, front, and back.

Record your ideas in a notebook or a method of your choice.

A rectangular prism with squares on 2 opposite ends (left and right ends).

Show Answer

Number of triangular sides	Number of rectangular sides	Number of square sides	Number of trapezoidal sides
0	4	2	0

When you are ready, press 'Show Answer' to reveal a solution.

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Number of triangular sides	Number of rectangular sides	Number of square sides	Number of trapezoidal sides
4	0	1	0

When you are ready, press 'Show Answer' to reveal a solution.

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Number of triangular sides	Number of rectangular sides	Number of square sides	Number of trapezoidal sides
2	3	0	0

When you are ready, press 'Show Answer' to reveal a solution.

Show Answer

Number of triangular sides	Number of rectangular sides	Number of square sides	Number of trapezoidal sides
0	4	0	2

Action

Task 1: Defining surface area

What is surface area?

Surface area refers to the total area of the surface of a three-dimensional object.

Why would someone want to measure the surface area of an object?

Consider this question and record your ideas in a notebook or a method of your choice.

When you are ready, press 'Show Answer' to reveal a solution.

Show Answer

Here is a list of a few situations where knowing the surface area of a three-dimensional object would be valuable:

knowing how much paint to use for a surface
when we want to wrap something
when we build things

What is the difference between area and surface area?

The key difference between these two terms is that the area is the measurement of a flat surface, or two-dimensional shape, while surface area is the measurement of all exposed surfaces or faces of a three-dimensional object.

Task 2: Calculating surface area

Brainstorm some ideas for how we could calculate the surface area of three-dimensional objects.

Consider how the areas of two-dimensional shapes are calculated.

To calculate the surface area of a three-dimensional object you will:

determine their two-dimensional faces
calculate the area of each two-dimensional face of the three-dimensional object
add all of the areas together

How can we ensure we have not missed any of the faces of a three-dimensional object?

We can use nets to check our thinking.

A net is a pattern that, when folded, creates a model of a three-dimensional object.

Surface area of a cube

For example, observe this net for a cube.

4 squares lined up with 1 square on both the left and right side of the square, second from the top.

A cube has six faces. If we folded up this net, it would become a cube. Take a moment to explore the interactive. At this time, don't worry about the volume button. Press 'Unfold Shape' to reveal an animation of the net.

The first step to calculating the surface area of a cube is to calculate the area of each face.

Then, we would need to add up all of the faces. Since squares have all equal sides, what could we do to make this process more efficient for us?

We only need to calculate the area of one face, and then we can multiply that number by 6 (the number of faces the cube has).

Therefore, the formula for calculating the surface area of a cube is:

Surface area = $6 a^{2}$ , where a represents the side length of each square face.

If the side lengths of the square in the net above are 12 cm, le's calculate what the surface area would be.

$S A = 6 a^{2}$

$S A = 6 \times (a \times a)$

$S A = 6 \times (12 c m \times 12 c m)$

$S A = 6 \times 144 c m^{2}$

$S A = 864 c m^{2}$

The surface area of this cube is $S A = 864 c m^{2} .$

Now let's go back to our Minds On activity. How would knowing the surface area of the box help us determine how much paint is needed to cover the surface of the box?

When you purchase paint, the can or container usually states on the label how much surface it will cover. That way you can compare the surface area of what you are painting to what the paint container will cover, and know how much paint to buy.

Would this formula be the same for all three-dimensional objects? Why, or why not? Record your ideas in a notebook or a method of your choice. Be prepared to share what you think.

When you are ready, press 'Show Answer' to reveal the answer.

Show Answer

The formula will not be the same for all three-dimensional objects, just like the formula for finding the area of two-dimensional shapes is not the same for all two-dimensional shapes. This is because not all three-dimensional objects have the same two-dimensional faces!

Le's explore other three-dimensional objects.

Surface area of a prism

A prism is a three-dimensional object that has two parallel and congruent bases. Prisms will be named differently based on the specific three-dimensional object. For example, a rectangle-based prism will be a prism that has two congruent rectangles as its base.

Surface area of rectangle-based prism

For example, observe this net of a rectangle-based prism.

Take a moment to explore the interactive. At this time, don't worry about the volume button. Press 'Unfold Shape' to reveal an animation of the net.

To calculate the surface area of a prism, you will add up all the areas of each face. Consider a rectangle-based prism that has a length of 10 cm, width of 3 cm, and height of 6 cm.

Rectangle-based prism with a length of 10 cm, width of 3 cm, and height of 6 cm.

Let's calculate the area of each face.

Base 1:

$A = 6 \times 3$

$= 18 c m^{2}$

Base 2:

$A = 6 \times 3$

$= 18 c m^{2}$

Rectangle 1:

$A = 10 \times 6$

$= 60 c m^{2}$

Rectangle 2:

$A = 10 \times 6$

$= 60 c m^{2}$

Rectangle 3:

$A = 10 \times 3$

$= 30 c m^{2}$

Rectangle 4:

$A = 10 \times 3$

$= 30 c m^{2}$

$SA = 18 + 18 + 60 + 60 + 30 + 30$

$SA = 216 c m^{2}$

Is there another way we could have recorded this?

We know that there are two of the same bases that have the same dimensions. We can record it in our formula like this:

$2 (b) = 2 (18)$
We know that there are two more rectangles with the same dimensions. We can record it in our formula like this:

$2 (rectangle dimensions) = 2 (60)$
Lastly, there are two more rectangles with the same dimensions. We can record it in our formula like this:

This way, our surface area formula would be:

$S A = 2 (18) + 2 (60) + 2 (30)$

$= 18 + 120 + 60 c m^{2}$

$= 216 c m^{2}$

Surface area of triangular prism

Now try calculating the surface area of this triangular prism on your own. Record your calculations in a notebook or a method of your choice. To help you find the surface area you may want a tip.

Press 'Hint' to access the tip.

Hint

Consider how to calculate the area of a triangle. Use the net or a physical object to help you identify each of the faces of the prism.

Triangular prism and net of length = 24 cm, width = 8 cm, height = 12 cm, triangle side = 10 cm

Take a moment to explore the interactive. At this time, don't worry about the volume button. Press 'Unfold Shape' to reveal an animation of the net.

Be prepared to share your answer.

When you are ready, press 'Show Answer' to reveal the answer.

Show Answer

The formula for the surface area of a triangular prism is:

$S A = b h + (b l + s l + s l)$

$S A = 8 \times 12 + (8 \times 24 + 10 \times 24)$

$S A = 96 + (192 + 240 + 240)$

$S A = 96 + 672$

$S A = 768 c m^{2}$

Surface area of a square-based pyramid

A pyramid is a three-dimensional object that has one polygon as its base. Triangles will be joined on each side of the base and will meet at the top of the triangle (the apex). The shape of the base is what gives a pyramid its name. For example, a pyramid that has a square as its base is called a square-based pyramid.

Here is a net of a square-based pyramid:

This square-based pyramid's height is 18 cm, length is 7 cm, and width is 7 cm. The height is the height of each triangle on the sides. To calculate the surface area of a pyramid, we will follow the same steps as for a prism or a cube.

Square-based pyramid with net that has a length of 7 cm, width of 7 cm, and height of 18 cm.

Take a moment to explore the interactive. At this time, don't worry about the volume button. Press 'Unfold Shape' to reveal an animation of the net.

Let’s calculate the area of each face.

Base:

$A = 7 \times 7$

$A = 49 c m^{2}$

Triangle 1:

$A = 7 \times 18 \div 2$

$A = 63 c m^{2}$

Triangle 2:

$A = (7 \times 18) \div 2$

$A = 63 c m^{2}$

Triangle 3:

$A = (7 \times 18) \div 2$

$A = 63 c m^{2}$

Triangle 4:

$A = (7 \times 18) \div 2$

$A = 63 c m^{2}$

$S A = 49 + 4 (63)$

$S A = 301 c m^{2}$

Surface area of a rectangle-based pyramid

Now try calculating the surface area of this pyramid.

Record your calculations in a notebook or a method of your choice.

Rectangle-based pyramid with net of length = 32 cm, width = 10 cm. Side heights are 13 cm and 20 cm

Press 'Hint' to reveal the tip.

Hint

Consider that this is a rectangle-based pyramid. The length of the rectangular base is 32 cm and the width is 10 cm. The height of the two triangles connected to the length of the base are 13 cm. The height of the two triangles connected to the width of the base are 20 cm. There are two different sized triangles since the base isn't a square.

Task 3: Surface area word problems

Answer any three of the five following questions about surface area.

Record your calculations in a notebook or a method of your choice.

Question 1

A company designed a new pencil case in the shape of a square-based pyramid. They will use paper to cover the pencil case before delivering it. How much paper did the company need to use for the pencil case?

Square-based pyramid that has a length of 8 cm, width of 8 cm, and height of 17 cm.

When you are ready, press 'Show Answer' to reveal the answer.

Show Answer

Base area = 64 cm²

Each triangle area = 68 cm²

Surface area = 4 × 68 + 64 = 336 cm²

Question 2

A local bakery is baking a cake for a community fundraiser. The cake measures 8 inches by 8 inches by 2 inches high. The bakers need to know how many square centimetres of icing are needed for the cake (note that they won't put icing on the bottom of the cake). How many square cm of icing are needed to cover the cake?

Un-iced chocolate rectangle cake 8 cm long by 8 cm wide by 2 cm tall with icing container beside it.

When you are ready, press 'Show Answer' to reveal the answer.

Show Answer

Top area = 64 cm²

Each side area = 16 cm²

Surface area = 4 × 16 + 64 = 128 cm² of icing

Question 3

Student A finds a box and wants to use it for an art project. The box is flat and needs to be put back together. It has the shape of a rectangle-based prism. If the length is 10 cm, the width is 4 cm, and the height is 15 cm, what is the surface area of the box?

When you are ready, press 'Show Answer' to reveal the answer.

Show Answer

The top and bottom area (each) = 40 cm²

Front and back (each) area = 150 cm²

Each side area = 60 cm²

Surface area = (2 ×40) + (2 × 150) + (2 × 60) = 500 cm²

Question 4

If the surface area of a rectangular prism is 900 cm², what could the dimensions of the prism be?

Rectangular prism with a surface area of 900 square centimetres with other measurements unknown.

When you are ready, press 'Show Answer' to reveal the answer.

Show Answer

Answers will vary. One possible rectangular prism is 10 cm × 12 cm × 15 cm

Top and bottom (each) areas = 180 cm²

Each side area = 150 cm²

Front and back (each) area = 120 cm²

Surface area = (2 × 180) + (2 × 150) + (2 × 120) = 900 cm²

Here is what it could look like.

Question 5

There are two boxes. The first box's length is 10 cm, width is 4 cm, and height is 7 cm. The second box's length is 8 cm, width is 5 cm, and height is 5 cm. If both these boxes were made from construction paper, would one of them be better for the environment? Justify your response.

When you are ready, press 'Show Answer' to reveal the answer.

Show Answer

First box

Surface area = (2 × 10 × 4) + (2 × 10 × 7) + (2 × 4 × 7)

Surface area = 80 + 140 + 56

Surface area = 276 cm²

Second box

Surface area = (4 × 8 × 5) + (2 × 5 × 5)

Surface area = 160 + 50

Surface area = 210 cm²

The second box might be better for the environment since it uses less cardboard.

Consolidation

Task 1: Making word problems

Now that you have had practice solving word problems about surface area, it is time to make your own!

Create three word problems that relate to surface area. After you have created your word problems, and solved them, share them with a partner, if possible.

Record your word problems and calculations in a notebook or a method of your choice.

Task 2: Questions to explore

Answer the following questions:

1) How would you explain calculating surface area to someone? What is most important for them to understand?

2) Do you think this is a valuable skill to have? Why or why not?

Record your ideas in a notebook or a method of your choice.

Reflection

As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.

I feel...

I have an excellent understanding of the concepts I have a very good understanding of the concepts I have some understanding but need to review some more my understanding of the concepts needs a lot more work

Now, record your ideas using a voice recorder, speech-to-text, or writing tool.

Connect with a TVO Mathify tutor

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Learning goals

Success criteria

Minds On

Inferring and observing shapes on sides

Action

Task 1: Defining surface area