Add face areas to find prism and pyramid surface area

Ontario Grade 6 Mathematics

Spatial Sense

Finding the surface area of prisms and pyramids

Imagine covering a closed box with paper. Each flat outside part of the box is a face. The amount of paper needed is the total area of all its faces. That total is called the surface area. A prism has two matching end faces and side faces. A pyramid has one base and triangular side faces. In this lesson, you will find surface area by adding face areas. You will use measurements stated in each example.

What you will learn

Explain that surface area is the total area of all the outside faces of a solid.
Find or use the area of each face of a prism or pyramid.
Add the face areas and give the answer in square units.
Check that every face has been counted once.

Start with faces and area

A face is one flat surface on a solid. A box has six faces. A pyramid has a base and triangular faces that meet at one point.

Area tells how much flat space a shape covers. For a rectangle, multiply its length by its width. For a triangle, multiply its base by its height and then divide by two. The triangle’s height is the straight distance from its base to the opposite point. Use the height given for that triangular face.

Area is measured in square units, such as square centimetres. Surface area is also measured in square units. Find each face area before adding the areas.

These area rules can be written as

A_{ ext{rectangle}}=lw

and

A_{ ext{triangle}}=\frac{bh}{2}

.

Count the outside faces.
Find each face’s area using its shape and given measurements.
Add the face areas and write the answer in square units.

Prism surface area

A prism has two matching faces at its ends. Its other faces connect the matching ends. A rectangular prism, such as a box, has six rectangular faces.

A net shows the faces laid flat. It can help you see which faces belong to a solid. For a rectangular prism, there are two faces of each of three sizes. Find the area of one face of each size, then add each size twice. This is the same as adding all six face areas.

Do not count only the faces that are easy to see in a drawing. A solid drawing may hide a face. Use the net or a face list to include every outside face exactly once.

SA=A_1+A_2+\\cdots+A_n

A rectangular prism has three pairs of matching faces.
A net can help keep track of all faces.
The total is the sum of the areas of every face.

Pyramid surface area

A pyramid has one base. Its side faces are triangles. The number and shape of the triangles depend on the base. For example, a pyramid with a square base has four triangular side faces.

Find the base area and the area of each triangular face. Then add those areas. If several faces have the same measurements, you may find one area and add it that many times. Make sure a triangle’s given height belongs to that triangular face; do not use a side length in its place.

A face list is useful when the triangles are not all the same size. Write one entry for the base and one for each triangle, then add the entries.

Include the base as well as every triangular side face.
Use the correct height for each triangle.
Repeated face areas may be added by repeating the same number.

A careful checking routine

Before adding, label each face area with its shape or place in the solid. After adding, check the face count against the solid: six faces for a rectangular prism, or one base plus all its triangular sides for a pyramid.

Keep units consistent. If the face measurements are in centimetres, the areas and surface area are in square centimetres. Do not add lengths to areas. A length measures a distance; an area measures a flat region.

The examples use stated, hypothetical measurements. They show how the addition works. They are not measurements of a real object.

Check face count and repeated faces.
Add areas, not side lengths.
Use square units in the final answer.

Worked example

A rectangular prism

A closed rectangular box has dimensions 6 cm by 4 cm by 3 cm. Find its surface area.

Rectangular prism net

Schematic net for a box with stated dimensions 6 cm, 4 cm, and 3 cm. The face areas occur in three matching pairs.

Find the three face sizes
The three different rectangles have side lengths 6 cm and 4 cm, 6 cm and 3 cm, and 4 cm and 3 cm. Each size appears on two opposite faces.
$6\times4=24,\quad6\times3=18,\quad4\times3=12$
Add all six areas
Add each face area twice because the box has two faces of each size. The result is in square centimetres.
$24+24+18+18+12+12=108$

Answer: The surface area is 108 cm².

Check: There are six face areas in the sum, and the answer uses square centimetres.

Worked example

A triangular prism

A triangular prism has two matching triangular ends. Each triangle has base 6 cm and height 4 cm. The three rectangular side faces measure 6 cm by 8 cm, 5 cm by 8 cm, and 5 cm by 8 cm. Find its surface area.

Find the triangular areas
Each triangular end has area half of 6 times 4. There are two matching ends, so include that area twice.
$\frac{6\times4}{2}=12$
Find the rectangular areas
The side-face areas are 48, 40, and 40 square centimetres. Add these to the two triangular areas.
$6\times8=48,\quad5\times8=40$
Add every face area
The two ends contribute 24 square centimetres. The three rectangles contribute 128 square centimetres. Together they make 152 square centimetres.
$12+12+48+40+40=152$

Answer: The surface area is 152 cm².

Check: The sum includes five faces: two triangles and three rectangles.

Worked example

A square-based pyramid

A pyramid has a square base with side length 5 cm. Each of its four triangular side faces has base 5 cm and height 4 cm. Find its surface area.

Find the base area
The base is a square, so multiply its side length by itself.
$5\times5=25$
Find the area of one triangle
Each triangular face has area half of its base times its height. The four triangles have the same stated measurements.
$\frac{5\times4}{2}=10$
Add the base and four triangles
The four triangles contribute 40 square centimetres. Add the square base area to get the total.
$25+10+10+10+10=65$

Answer: The surface area is 65 cm².

Check: The total includes one square base and four triangular faces.

Common mistakes and how to avoid them

Adding only the faces visible in a solid drawing.

Correction: List every outside face, including hidden faces, or use a net.

Forgetting to include the base of a pyramid.

Correction: Add the base area and all triangular side-face areas.

Adding side lengths instead of face areas.

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

Writing a surface-area answer in centimetres.

Correction: Area uses square units, such as cm².

Lesson summary

Surface area is the total area of all the outside faces.
Find each face area, then add every face once.
A prism has two matching ends and connecting side faces.
A pyramid has a base and triangular side faces.
Give the final answer in square units.

Check your understanding

Question 1

A rectangular prism has three pairs of matching faces. The area of one face in each pair is 7 cm², 9 cm², and 11 cm². What is its surface area?

27 cm²
54 cm²
63 cm²
27 cm

Show answer and explanation

54 cm²

Each of the three areas appears twice, so add 7 + 7 + 9 + 9 + 11 + 11 to get 54 cm².

Question 2

A pyramid has a base area of 18 cm² and three triangular faces, each with area 6 cm². What is its surface area?

24 cm²
36 cm²
18 cm²
36 cm

Show answer and explanation

36 cm²

Include the base and all three triangles: 18 + 6 + 6 + 6 = 36 cm².

Question 3

A prism’s face areas are 12 cm², 12 cm², 8 cm², 8 cm², 5 cm², and 5 cm². What should the final unit be?

cm
cm²
cm³
No unit is needed

Show answer and explanation

cm²

The listed values are areas, so their total is written in square centimetres, cm².

Key terms

Face: A flat outside surface of a solid.
Area: The amount of flat space a shape covers.
Surface area: The sum of the areas of all the outside faces of a solid.
Prism: A solid with two matching end faces and side faces connecting them.
Pyramid: A solid with one base and triangular side faces.
Net: A drawing that shows the faces of a solid laid flat.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation E2.6. 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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E2.6 · Add face areas to find prism and pyramid surface area

Ontario Grade 6 Mathematics

Finding the surface area of prisms and pyramids

What you will learn

Start with faces and area

Prism surface area

Pyramid surface area

A careful checking routine

A rectangular prism

A triangular prism

A square-based pyramid

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 6 Math

About this lesson and its review