Connect prism and pyramid nets with their surface areas

Ontario Grade 6 Mathematics

Spatial Sense

Seeing every face of a prism or pyramid

Imagine opening a small box along some edges and laying it flat. The flat shape is called a net. It shows the box’s faces before they are folded back into a solid. Surface area is the total area of all the outside faces. In this lesson, you will use nets to see which faces to measure and why their areas must be added. The measurements in the examples are hypothetical.

What you will learn

Explain how a net shows the faces on the outside of a solid.
Connect the area of each face in a net to the solid’s surface area.
Find surface area by adding the areas of all the faces shown in a net.

A net shows the outside faces

A face is a flat surface of a solid. A prism has two matching end faces and side faces between them. A pyramid has one base and triangular side faces that meet at a point.

A net is a flat arrangement of all the faces of a solid. When its edges are folded, the faces join to make the solid. A good net has each face the solid needs, with no extra face and no missing face.

Area tells how much flat space a face covers. You may already know that the area of a rectangle is its length multiplied by its width. For example, a rectangle that is

4\text{ cm}

long and

3\text{ cm}

wide has area

12\text{ cm}^2

. The small 2 means square centimetres.

Surface area is the total area of all the faces on the outside of a solid. To connect a net to surface area, find the area of each face in the net and add those areas. The net helps you count each outside face once.

\text{surface area}

=sum of the areas of all faces

A net is a flat layout of a solid’s faces.
Surface area is the sum of the areas of all outside faces.
Use square units, such as $\text{cm}^2$ , for area.

Read a prism net

A rectangular prism has six rectangular faces. Opposite faces match in size. Its net can show two end rectangles and four side rectangles. The side rectangles wrap around the four edges of an end face.

For a rectangular face, multiply its length by its width. Then add the areas of all six faces shown in the net. If two faces match, you can find the area of one and add it twice. This is a way to organize the addition, not a different meaning of surface area.

A net picture can be schematic, which means it shows how the faces connect but may not be drawn to exact scale. Use the stated measurements to find the areas.

\text{rectangle area}=\text{length}\times\text{width}

Count every face in the net.
Find each face’s area using its side measurements.
Add the face areas, keeping the same square unit.

Read a pyramid net

A square-based pyramid has one square base and four triangular side faces. Its net has the square with one triangle attached to each of its four sides. Folding the triangles up brings them together at the pyramid’s point.

A triangle’s area is half the area of a rectangle with the same base and height. The triangle’s height is the straight distance from its base to its opposite point. Once you know the area of the base and of each triangle, add all five face areas.

The net makes it clear that the base is part of the outside surface too. Do not leave it out just because it rests on a table in a model.

\text{triangle area}=\frac{\text{base}\times\text{height}}{2}

A square-based pyramid net has one square and four triangles.
Include the base and all side faces.
Use matching measurements to find the area of each face.

A careful plan

Before calculating, name the solid and count its faces in the net. Match each face to the measurements that belong to it. Find each face area, then add the results.

Check that the final unit is a square unit. Also check that the total is larger than the area of any one face, since it includes all the faces. In the examples, the measurements are hypothetical and are provided only to practise connecting nets with surface area.

Count, measure, find face areas, and add.
Use the net to make sure no face is missed or counted twice.
Write the answer in square units.

Worked example

Surface area of a rectangular prism

A hypothetical rectangular prism has length

5\text{ cm}

, width

3\text{ cm}

, and height

2\text{ cm}

. Use its net to find its surface area.

Rectangular prism net

Schematic net for the prism with length 5 cm, width 3 cm, and height 2 cm. The net is not to scale.

Count the faces
The net of a rectangular prism has six rectangles. There are two faces of each size: $5\text{ cm}$ by $3\text{ cm}$ , $5\text{ cm}$ by $2\text{ cm}$ , and $3\text{ cm}$ by $2\text{ cm}$ .
Find and add the face areas
Multiply the side lengths for each rectangle. Add two of each area because each rectangle size appears twice in the net.
$2(5\times3)+2(5\times2)+2(3\times2)=62\text{ cm}^2$
State the surface area
The total area of the six outside faces is $62$ square centimetres.

Answer:

62\text{ cm}^2

Check: The three different face areas are

15\text{ cm}^2

,

10\text{ cm}^2

, and

6\text{ cm}^2

. Two of each gives

30+20+12=62\text{ cm}^2

.

Worked example

Surface area of a square-based pyramid

A hypothetical square-based pyramid has a base with side length

4\text{ cm}

. Each triangular face has base

4\text{ cm}

and height

5\text{ cm}

. Use its net to find its surface area.

Find the base area
The base is a square. A square is a rectangle with equal side lengths, so multiply $4\text{ cm}$ by $4\text{ cm}$ .
$4\times4=16\text{ cm}^2$
Find the area of one triangle
Multiply the triangle’s base by its height, then take half. This gives the area of one triangular face.
$(4\times5)\div2=10\text{ cm}^2$
Add all five face areas
The net has four matching triangles and one square base. Add the base area to the areas of all four triangles.
$16+4\times10=56\text{ cm}^2$

Answer:

56\text{ cm}^2

Check: The four triangles together have area

40\text{ cm}^2

. Adding the

16\text{ cm}^2

square base gives

56\text{ cm}^2

.

Worked example

Use a net to check a total

A hypothetical prism net has two matching end faces, each with area

14\text{ cm}^2

. Its four side faces have areas

6\text{ cm}^2

,

6\text{ cm}^2

,

9\text{ cm}^2

, and

9\text{ cm}^2

. Find the surface area.

Group matching faces
The net lists two end faces with the same area. The side faces make two more matching pairs, one pair of area $6\text{ cm}^2$ and one pair of area $9\text{ cm}^2$ .
Add all six areas
Add both end faces and each of the four side faces. Grouping equal areas helps keep the count clear.
$2\times14+2\times6+2\times9=58\text{ cm}^2$
Connect the sum to the solid
The net shows six outside faces in total. Their combined area is the prism’s surface area.

Answer:

58\text{ cm}^2

Check: Adding one face at a time gives

14+14+6+6+9+9=58\text{ cm}^2

.

Common mistakes and how to avoid them

Leaving out the base of a pyramid.

Correction: The base is an outside face, so include its area along with the triangular faces.

Adding side lengths instead of face areas.

Correction: Find the area of each face first. Then add those areas.

Counting only one face from a matching pair.

Correction: Check the net for every face. If two faces match, include both areas.

Writing a surface area in centimetres instead of square centimetres.

Correction: Area measures flat space, so use square units such as

\text{cm}^2

.

Lesson summary

A net lays out the faces of a solid.
A prism or pyramid’s surface area is the sum of the areas of all its outside faces.
Use the net to count faces, find each face area, and add the areas.
Write the total in square units.

Check your understanding

Question 1

A net has one square base with area

25\text{ cm}^2

and four triangles, each with area

8\text{ cm}^2

. What is the pyramid’s surface area?

$33\text{ cm}^2$
$57\text{ cm}^2$
$32\text{ cm}^2$
$65\text{ cm}^2$

Show answer and explanation

57\text{ cm}^2

Include the base and all four triangles:

25+4\times8=57\text{ cm}^2

.

Question 2

A prism net shows six faces with areas

10\text{ cm}^2

,

10\text{ cm}^2

,

7\text{ cm}^2

,

7\text{ cm}^2

,

4\text{ cm}^2

, and

4\text{ cm}^2

. What is the surface area?

$21\text{ cm}^2$
$32\text{ cm}^2$
$42\text{ cm}^2$
$84\text{ cm}^2$

Show answer and explanation

42\text{ cm}^2

Add all six face areas:

10+10+7+7+4+4=42\text{ cm}^2

.

Key terms

Face: A flat surface of a solid.
Net: A flat layout of the faces of a solid that can be folded to make the solid.
Prism: A solid with two matching end faces and side faces joining them.
Pyramid: A solid with one base and triangular side faces that meet at a point.
Surface area: The total area of all the outside faces of a solid.
Square unit: A unit for area, such as a square centimetre, written $\text{cm}^2$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation E2.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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E2.5 · Connect prism and pyramid nets with their surface areas

Ontario Grade 6 Mathematics

Seeing every face of a prism or pyramid

What you will learn

A net shows the outside faces

Read a prism net

Read a pyramid net

A careful plan

Surface area of a rectangular prism

Surface area of a square-based pyramid

Use a net to check a total

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through Grade 6 Math

About this lesson and its review