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

Surface Area of Prisms and Pyramids

E2.6 A1About 45 minutes

Learning goals

I can find the surface area of a rectangular prism or a cube by adding the areas of its faces.
I can find the surface area of a triangular prism and a square-based pyramid.
I can solve problems about wrapping, painting or covering objects.

Words to know

surface area
The total area of all the faces of a 3D object, measured in square units like cm² or m².

face
A flat surface of a 3D object.

dimensions
The measurements of an object, such as its length, width and height.

triangle height (on a pyramid)
The height of one triangle face, measured from the middle of its base straight up the face to the top point. It is longer than the height of the pyramid itself.

Let's start with a gift

It's the holidays, and you're wrapping a box for your cousin. How much paper do you need, at the very least, to cover the box with no overlaps? You need the surface area: the total area of all its faces.

In the last lesson, you saw that a net is all the faces laid flat. So: surface area = the area of the net = the sum of the areas of all the faces.

Wrapped gift boxes of different sizes on a table with a roll of wrapping paper, scissors and tape. — The least paper that can cover a box with no overlap equals its surface area, the total area of all its faces. AI-generated image

The net of a box 5 cm long, 3 cm wide and 2 cm tall, drawn in proportion. The numbers on the faces are their areas: two faces of 15 cm², two of 10 cm² and two of 6 cm². The total is 62 cm².

Worked example

Example 1: A rectangular prism

Find the surface area of a box that is 5 cm long, 3 cm wide and 2 cm tall.

A rectangular prism has 3 pairs of matching faces.
Top and bottom: 5 × 3 = 15 cm² each. 2 × 15 = 30 cm².
Front and back: 5 × 2 = 10 cm² each. 2 × 10 = 20 cm².
Two ends: 3 × 2 = 6 cm² each. 2 × 6 = 12 cm².
Add: 30 + 20 + 12 = 62 cm².

62 cm²

Organize your work

With so many faces, it's easy to miss one or count one twice. A table helps:

List each kind of face.
Write its area and how many there are.
Multiply and add.

For a cube, all 6 faces are the same square: surface area = 6 × (edge × edge).

The net of a triangular prism whose bases are right triangles with sides of 3 cm, 4 cm and 5 cm, and whose length is 6 cm, drawn in proportion. The rectangles have areas of 18, 24 and 30 cm², and each triangle has an area of 6 cm².

Worked example

Example 2: A triangular prism

A triangular prism has right-triangle bases with sides 3 cm, 4 cm and 5 cm. The prism is 6 cm long. Find its surface area.

Rectangles: one for each side of the triangle, each 6 cm long: 3 × 6 = 18, 4 × 6 = 24, 5 × 6 = 30. Total 72 cm².
Triangles: the two legs (3 cm and 4 cm) meet at the right angle, so use them as base and height: 3 × 4 ÷ 2 = 6 cm². Two of them: 12 cm².
Add: 72 + 12 = 84 cm².

84 cm²

Face	Area of one	How many	Total
Rectangle 3 cm × 6 cm	18 cm²	1	18 cm²
Rectangle 4 cm × 6 cm	24 cm²	1	24 cm²
Rectangle 5 cm × 6 cm	30 cm²	1	30 cm²
Triangle (base 3 cm, height 4 cm)	6 cm²	2	12 cm²
Surface area			84 cm²

A table keeps track of every face of the triangular prism in Example 2.

Worked example

Example 3: A square-based pyramid

A square-based pyramid has a base with 4 cm sides. Each triangle face has a base of 4 cm and a height of 6 cm. Find its surface area.

Base: 4 × 4 = 16 cm².
Each triangle: 4 × 6 ÷ 2 = 12 cm². There are 4: 4 × 12 = 48 cm².
Add: 16 + 48 = 64 cm².

64 cm²

Just for fun

Wrapping a gift with the least paper is a surface-area problem. Wrapping it with a whole roll of tape is just a sticky situation.

Check your units

Surface area is an area, so the answer is always in square units (cm², m²). Before you add, make sure all the measurements are in the same unit. If one is in metres and the others are in centimetres, convert first (you learned how in the metric units lesson).

Hands-on activity

Wrap it up

You need: 2 or 3 boxes from the recycling bin (a cereal box, a tissue box, a toothpaste box), a ruler, a pencil and paper.

Measure the length, width and height of each box to the nearest centimetre.
Make a table for each box: the kind of face, the area of one, how many, and the total.
Find each box's surface area. Which box needs the most wrapping paper?
Estimate first, then check: before measuring the next box, guess its surface area. How close were you?
Bonus: find a box shaped like a triangular prism (some snack or gift boxes are). Measure it and find its surface area.

Math talk: "Two boxes have about the same size. Why might one need more paper?"

Your turn

Find the surface area of a cube with 5 cm edges.

Find the surface area of a box that is 6 cm by 4 cm by 3 cm.

A square-based pyramid has a base with 10 cm sides. Each triangle face has a height of 8 cm. Find its surface area.

A triangular prism has right-triangle bases with sides 6 cm, 8 cm and 10 cm. It is 10 cm long. Find its surface area.

Why is surface area measured in square units?

A gift box is 30 cm by 20 cm by 10 cm. What is the least amount of wrapping paper that could cover it?

More practice: warm-up

How many faces does a rectangular prism have?

What is the area of a 5 cm by 4 cm face?

What is the surface area of a cube with 1 cm edges?

Find the area of a triangle with a base of 6 cm and a height of 4 cm.

Is surface area measured in cm, cm² or cm³?

More practice: on your own

Find the surface area of a box that is 10 cm by 5 cm by 2 cm.

A cube has a surface area of 54 cm². How long is each edge?

A square-based pyramid has a base with 8 m sides, and each triangle face has a height of 10 m. Find its surface area.

A triangular prism is 10 cm long. Its bases are triangles with sides of 5 cm, 5 cm and 6 cm; the triangle's height on the 6 cm side is 4 cm. Find its surface area.

Which needs more paper to cover: a cube with 4 cm edges, or a box 8 cm by 4 cm by 2 cm? (They hold the same amount.)

Challenge

A box with no lid is 20 cm long, 15 cm wide and 10 cm tall. How much cardboard is needed to make it?

Two cubes with 2 cm edges are glued face to face to make a 4 cm by 2 cm by 2 cm prism. Compare the surface area of the prism with the total surface area of the two separate cubes. Explain the difference.

A triangular prism has right-triangle bases with sides 3 cm, 4 cm and 5 cm. Its surface area is 132 cm². How long is the prism?

Check your understanding

Quiz

What is the surface area of a cube with 3 cm edges?

A box is 4 cm by 3 cm by 1 cm. What is its surface area?

How many triangle faces does a square-based pyramid have?

A square-based pyramid has a 6 cm by 6 cm base and triangle faces with a height of 4 cm. What is its surface area?

Which unit could be used for surface area?

A triangular prism's two triangles have a total area of 20 cm², and its three rectangles have a total area of 150 cm². What is its surface area?

For parents and teachers

Big idea: The Ontario curriculum (E2.6) asks Grade 6 students to determine the surface areas of prisms and pyramids by calculating the areas of their two-dimensional faces and adding them together. It builds directly on nets (E2.5) and on the area formulas for rectangles and triangles.

Common difficulties:

Missing or double-counting faces. A table (kind of face, area, how many, total) solves most errors.
Using the pyramid's height instead of the triangle's height. The triangle height runs up the slanted face.
Wrong units. Surface area is in square units (cm², m²).

Social-emotional learning (A1): These problems have several steps. Encourage your child to organize their work and to estimate first so they can tell whether an answer is reasonable.

How to help: Measure boxes from the recycling bin together and compare surface areas. When wrapping a present, ask "About how much paper will we need?"