Mathematics · Grade 7 · Surface Area and Volume · Term 3

Surface Area of Prisms

Calculating the total surface area of rectangular and triangular prisms using nets and formulas.

• Ontario Curriculum Expectations
• 7.G.B.6

About This Topic

Volume of right prisms in Grade 7 focuses on the concept of 'accumulation.' Instead of just memorizing V = lwh, the Ontario curriculum encourages students to think of volume as the area of the base multiplied by the height (V = Abase × h). This understanding allows students to calculate the volume of any right prism, including triangular and pentagonal ones, by seeing them as 'stacks' of 2D layers. This topic is a key part of the 'Measurement' strand and is essential for practical tasks like determining tank capacity or shipping costs.

Students explore the relationship between different units of measure, such as cubic centimeters and milliliters, which is vital for science and cooking. They also investigate how the volume of a triangular prism is exactly half that of a rectangular prism with the same base and height. This topic particularly benefits from hands-on, student-centered approaches where students can physically fill or build shapes to see the layers. Students grasp this concept faster through structured discussion and peer explanation.

Key Questions

1. Justify why a manufacturer might want to minimize the surface area of a package while keeping the volume the same.
2. Differentiate between calculating lateral area and total surface area.
3. Analyze how the dimensions of a prism affect its surface area.

Learning Objectives

• Calculate the surface area of rectangular prisms using nets and formulas.
• Calculate the surface area of triangular prisms using nets and formulas.
• Differentiate between lateral surface area and total surface area for prisms.
• Justify how changes in prism dimensions impact its total surface area.
• Analyze the relationship between a package's surface area and volume to explain manufacturing cost considerations.

Before You Start

Area of Rectangles and Triangles

Why: Students must be able to calculate the area of the basic 2D shapes that form the faces of prisms.

Volume of Rectangular Prisms

Why: Understanding volume as base area times height provides a foundation for understanding surface area as the sum of face areas.

Key Vocabulary

Surface Area	The total area of all the faces of a three-dimensional object. It represents the amount of material needed to cover the object's exterior.
Net	A two-dimensional pattern that can be folded to form a three-dimensional shape. For prisms, nets show all the rectangular and triangular faces laid out flat.
Lateral Surface Area	The sum of the areas of the sides of a prism, excluding the areas of the two bases. It is the 'wrapping' area of the prism.
Rectangular Prism	A prism with six rectangular faces. Its volume is calculated as length × width × height.
Triangular Prism	A prism with two triangular bases and three rectangular sides. Its volume is calculated as the area of the triangular base × height.

Watch Out for These Misconceptions

Common Misconception

Volume is only for rectangular boxes.

What to Teach Instead

Students often think the formula V = lwh is the only way to find volume. Teaching the 'Base Area x Height' method through active learning helps them apply volume concepts to any prism shape, including triangles.

Common Misconception

If you double the height, the volume quadruples.

What to Teach Instead

Students sometimes confuse volume scaling with area scaling. Building models with cubes helps them see that doubling only one dimension (height) simply doubles the number of layers, thus doubling the volume.

Active Learning Ideas

Inquiry Circle: Layer by Layer

Students use 1cm cubes to build a rectangular prism. They first build the base layer, calculate its area, and then 'stack' layers to reach a certain height, recording the total volume at each step to see the V = Base x Height relationship.

30 min·Small Groups

Simulation Game: The Water Displacement Challenge

Using graduated cylinders and various small prisms, students predict the volume of each object, then submerge them in water to measure the displacement. They compare their mathematical calculations to the physical results.

40 min·Pairs

Think-Pair-Share: Triangular vs Rectangular

Show a rectangular prism and a triangular prism with the same base dimensions and height. Students must predict the relationship between their volumes and then use 'sand pouring' to test if the triangular one is indeed half.

20 min·Pairs

Real-World Connections

• Packaging designers at companies like Canada Post use surface area calculations to determine the amount of cardboard needed for shipping boxes, directly impacting material costs and shipping weight.
• Architects and engineers consider the surface area of buildings when calculating heat loss or gain, influencing decisions about insulation and exterior materials.
• Manufacturers of canned goods, such as Maple Leaf Foods, analyze the surface area to volume ratio of their product containers to optimize material usage and shelf space efficiency.

Assessment Ideas

Quick Check

Provide students with a diagram of a rectangular prism (e.g., 5 cm x 3 cm x 2 cm). Ask them to calculate the total surface area and the lateral surface area, showing their work using both the net and formula methods.

Discussion Prompt

Pose the question: 'Imagine you have two boxes, one large and one small, that can hold the exact same volume of cereal. Which box likely uses less cardboard to make, and why?' Guide students to discuss the surface area to volume ratio.

Exit Ticket

Give students a net of a triangular prism. Ask them to calculate its total surface area. Then, ask them to write one sentence explaining why a company might want to minimize the surface area of a product package while keeping the volume constant.

Frequently Asked Questions

What is the general formula for the volume of a prism?

The general formula is Volume = Area of the Base × Height. This works for any right prism, whether the base is a rectangle, a triangle, or any other polygon.

Why is volume measured in cubic units?

Volume measures three-dimensional space (length, width, and height). A cubic unit (like cm³) represents a small cube that is 1 unit on all sides, which is the standard way to fill and measure 3D space.

How can active learning help students understand volume?

Active learning, such as stacking cubes or using water displacement, makes the concept of 'filling space' visible. When students build a prism layer by layer, they understand why we multiply the base area by the height, making the formula a logical conclusion rather than a rule.

How does volume relate to capacity (litres)?

In the metric system used in Canada, volume and capacity are directly linked: 1 cm³ is exactly equal to 1 mL. This makes it easy for students to convert their geometric measurements into liquid volumes for science or cooking.