Surface Area Cylinders

Overview

In this fifth lesson of Similarity to Solid Design, students connect a cylinder to its net and use the net to derive and apply the surface-area formula. They build on prior work with circles, circumference, radius, diameter, and area.

Learning intentions

Students will:

• connect a cylinder to its two-dimensional net;
• explain why the curved surface unwraps to a rectangle;
• determine the surface area of a cylinder using (SA=2\pi r^2+2\pi rh);
• communicate solutions using correct units and mathematical reasoning.

Success criteria

• I can identify the radius, diameter, and height of a cylinder.
• I can label the two circular bases and the rectangular curved surface on a net.
• I can calculate circular area and curved-surface area accurately.
• I can explain why the rectangle’s length is the cylinder’s circumference and report answers in square units.

Curriculum links

• Alberta Mathematics 10–12 Program of Studies: measurement and use of formulas to solve problems involving three-dimensional objects.
• Alberta Mathematics 10–12 Program of Studies: spatial sense, visualization, and connections between two-dimensional representations and three-dimensional objects.
• Alberta mathematics program emphasis: problem solving, mathematical reasoning, communication, and connections to real-world applications.
• Alberta Mathematics Kindergarten to Grade 12 Scope and Sequence: progression from circle measurement and formulas to surface area of three-dimensional objects.

Lesson structure (30 minutes)

1. 0–3 min · Warm-up: radius or diameter? Teacher displays a circle and asks students to identify its radius and diameter using the opening circle diagram. Students sketch a circle, mark both measures, and state the relationship (d=2r). Quickly address the common error of using diameter as radius.

2. 3–11 min · Mini-lesson: unwrap the cylinder. Teacher shows a cylinder and its net using the cylinder-net sequence and asks students to predict what each part represents. Students observe that the net contains two congruent circles and one rectangle. Establish:

• two bases: (2\pi r^2);
• curved surface: rectangle area (length\times height=(2\pi r)h=2\pi rh);
• total surface area: (SA=2\pi r^2+2\pi rh).

Model (r=3\text{ cm}), (h=5\text{ cm}): (SA=2\pi(3^2)+2\pi(3)(5)=48\pi\approx150.8\text{ cm}^2). Emphasize that surface area is measured in square units.

3. 11–18 min · Guided practice: label the net. Teacher distributes the Cylinder Surface Area worksheet and provides cylinder templates for students to inspect. On the displayed net in the guided-labelling slide, students label the two bases, radius, height, rectangle length, and rectangle width. In pairs, they calculate the area of one circle, the area of both circles, the rectangle’s area, and the total surface area. Check after each stage and ask: “Which measurement repeats around the cylinder?”

4. 18–26 min · Independent practice: apply the formula. Teacher directs students to complete the remaining questions on the Cylinder Surface Area worksheet, using (\pi\approx3.14) unless otherwise directed. Students show substitution, calculations, and units. Circulate and check that students use radius rather than diameter, include both bases, and distinguish linear from square units. Early finishers compare cylinders with the same height but different radii and describe how changing (r) affects surface area.

5. 26–28 min · Formative explanation check. Teacher displays the prompt in the explain-your-thinking slide: “Why is the rectangle’s length equal to the circumference?” Students explain to a partner using the sentence frame: “When the curved surface is unwrapped, its length is ___ because ___.” Invite two students to share; listen for (2\pi r).

6. 28–30 min · Closure and exit ticket. Teacher displays the summary and exit-ticket slide. Students complete: “A cylinder has diameter (10\text{ cm}). What is its radius? State the first part of the surface-area formula you would use.” Expected radius: (5\text{ cm}). Collect responses to identify students needing a review of diameter-to-radius conversion.

Resources

• the cylinder surface-area slide deck
• the Cylinder Surface Area worksheet
• Cylinder templates or nets
• Rulers
• Calculators
• Board or document camera
• Formula cards for students who need a scaffold

Assessment

• During the warm-up, check whether students distinguish radius from diameter.
• During guided practice, question students about the net and inspect whether they include both circular bases and use circumference for the rectangle’s length.
• Use the exit ticket and worksheet responses to identify errors with radius, the formula, calculator use, or square units.

Differentiation

• Support students with a formula card showing (d=2r), (C=2\pi r), circle area (=\pi r^2), and (SA=2\pi r^2+2\pi rh). Provide a labelled example net and allow calculator use.
• Read directions aloud, chunk the worksheet into one calculation at a time, and provide sentence frames for explanations. Pair students strategically and use clear visual models for EAL learners.
• For students requiring additional support, use a concrete cylinder template and physically trace the circumference as the rectangle’s length.
• Extend confident students by comparing cylinders with equal height but different radii and explaining which dimensions contribute most to the change in surface area.