Materials Needed

• Scientific calculator (with a $\pi$ button)
• Metric ruler or measuring tape
• At least one cylindrical object (e.g., a soup can, Pringles tube, or soda can)
• At least one spherical object (e.g., a tennis ball, orange, or basketball)
• Paper, pencils, and colored markers
• Optional: String (for measuring circumference)

Learning Objectives

By the end of this lesson, you will be able to:

• Calculate the Volume and Surface Area of cylinders.
• Calculate the Volume and Surface Area of spheres.
• Apply these formulas to solve real-world design and packaging problems.

1. The Hook: The "Best Value" Challenge (5 Minutes)

Scenario: You are a product designer for a new soda company. You have two choices for your packaging: a tall, skinny cylinder or a perfectly spherical "bubble" bottle. Both look cool, but one might cost way more to manufacture because it uses more plastic, while the other might hold less soda. How do you know which is better?

Discussion: If we want to know how much soda fits inside, are we looking for Surface Area or Volume? If we want to know how much plastic is needed for the bottle, which measurement matters then?

2. "I Do": Decoding the Cylinder (10 Minutes)

Think of a cylinder as a stack of circles. To find the space inside, we just need to know the area of one circle and how high the stack goes.

The Formulas:

• Volume ($V$): Area of the base $\times$ height $\rightarrow V = \pi r^2 h$
• Surface Area ($SA$): Think of the "Net." You have two circles (top and bottom) and a rectangle (the label).
  • Area of the two circles = $2 \times \pi r^2$
  • Area of the side (rectangle) = Circumference $\times$ height = $2 \pi r h$
  • Total $SA = 2\pi r^2 + 2\pi rh$

Example Walkthrough: Let's look at a standard soda can ($r = 3$cm, $h = 12$cm).

1. $V = \pi \times 3^2 \times 12 = \pi \times 9 \times 12 \approx 339.29 \text{ cm}^3$
2. $SA = (2 \times \pi \times 3^2) + (2 \times \pi \times 3 \times 12) \approx 56.55 + 226.19 = 282.74 \text{ cm}^2$

3. "We Do": Solving the Sphere (10 Minutes)

Spheres are unique because they only have one dimension: the radius ($r$). Everything depends on how far it is from the center to the edge.

The Formulas:

• Volume ($V$): $V = \frac{4}{3} \pi r^3$ (Watch out: that's "radius cubed"!)
• Surface Area ($SA$): $SA = 4 \pi r^2$ (Fun fact: the surface area of a sphere is exactly 4 times the area of its middle circle!)

Guided Practice: Let's calculate a tennis ball with a radius of $3.3$cm.

• Step 1 (Volume): $\frac{4}{3} \times \pi \times 3.3^3$. Calculation: $3.3 \times 3.3 \times 3.3 = 35.937$. Then multiply by $\pi$ and $4/3$. What do you get? (Target: $\approx 150.53 \text{ cm}^3$)
• Step 2 (Surface Area): $4 \times \pi \times 3.3^2$. Calculation: $3.3 \times 3.3 = 10.89$. Then multiply by $4$ and $\pi$. Result? (Target: $\approx 136.85 \text{ cm}^2$)

4. "You Do": The Real-World Measurement Lab (20 Minutes)

Now it’s your turn to be the engineer. Grab your physical objects and your ruler.

Task A: Measure and Calculate

1. Cylinder: Measure the height and the diameter (divide diameter by 2 to get the radius). Calculate the Volume and Surface Area.
2. Sphere: If it's hard to measure the radius directly, wrap a string around the widest part to get the circumference. Divide circumference by $2\pi$ to find the radius. Then calculate Volume and Surface Area.

Task B: The "Shipping Challenge"

You need to ship your sphere inside a cylinder-shaped tube that fits it perfectly (the height of the tube is equal to the diameter of the sphere, and the radius is the same).

• Calculate the volume of the empty space left in the tube after the sphere is dropped inside.
• Hint: Empty Space = Cylinder Volume - Sphere Volume.

5. Differentiation & Extensions

• Support: Use a "Formula Cheat Sheet" with a step-by-step calculator guide (e.g., "First, hit the radius button, then the $x^2$ button...").
• Challenge: "The Hemisphere Problem." If you cut your sphere exactly in half, what is the new Surface Area? (Note: Don't forget the new flat circular face created by the cut!)

6. Conclusion: Closure & Recap (8 Minutes)

Recap:

• The volume of a cylinder is just the base area ($\pi r^2$) times height ($h$).
• A sphere's surface area is exactly $4\pi r^2$.
• When calculating volume, units are always cubed (e.g., $\text{cm}^3$); for surface area, they are squared (e.g., $\text{cm}^2$).

Exit Ticket Question: If you double the radius of a cylinder but keep the height the same, does the volume double, or does it increase by more? Why? (Answer: It quadruples because the radius is squared!)

Success Criteria

You know you’ve nailed this lesson if you can:

1. Correctly identify the radius and height from a physical object or diagram.
2. Select the correct formula for either Volume or Surface Area.
3. Show your working steps clearly, including the units of measurement.
4. Explain the difference between the "space inside" (Volume) and the "material outside" (Surface Area).