LESSON PLAN

Volume of Prisms and Cylinders

Apothem Team

Grade 7 · Measurement

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Show a soup can and a cracker box: "Both claim 500 mL. How would you check WITHOUT opening them?" Proposals converge on computing the space inside — and the box surrenders to last year's $V = A_{\text{base}} \times h$ , but the can's circular base is new territory.

The punchline of the unit: the SAME extrusion formula covers both — any prism or cylinder is a base area swept through a height.

Explore

Extrusion lab: build "cylinders" by stacking identical circular chips and prisms by stacking congruent cardboard cross-sections — the stack's volume is (one layer's area) × (number of layers), no matter the layer's shape.

Then the verification station: compute a real can's volume from measurements (diameter and height with a ruler), then fill it with water into a measuring cup. Typical result: computed 412 cm³, measured ≈ 410 mL — the 1 cm³ = 1 mL bridge confirmed by experiment, with measurement error discussed like scientists.

Formalize

Formalize the universal prism/cylinder formula and the cylinder's specialization:

$V = A_{\text{base}} \times h \qquad V_{\text{cyl}} = \pi r^2 h$

The radius ritual carries over from the circles unit: write $r$ explicitly before computing (cans are measured by diameter; the formula eats radius). Exact answers keep $\pi$ ; applied answers convert with the 1 cm³ = 1 mL bridge when liquids enter.

Practice

Practice: one rectangular prism (review), two cylinders (one given diameter), one triangular prism; one reverse problem (a cylinder holds 1 L with radius 5 cm — height?); one comparison ("which mug holds more").

Exit ticket: a can has radius 4 cm, height 10 cm. Exact and approximate volume, and the mL it holds. ( $160\pi \approx 502.7$ cm³ ≈ 503 mL.)

Exit ticket

Exit ticket: a can has radius 4 cm, height 10 cm. Exact and approximate volume, and the mL it holds. ( $160\pi \approx 502.7$ cm³ ≈ 503 mL.)

TIP Keep a 1-litre cube (10 cm × 10 cm × 10 cm) model on the desk all week — the cm³↔mL↔L chain stops being abstract when the litre is a visible box.

WORKED EXAMPLES

Example 1 — The rain barrel: cylinder volume at yard scale

The barrel: diameter 60 cm, height 90 cm. How many litres does it hold?

Step 1: Radius first: $r = 30$ cm.

Step 2: Base area: $\pi(30)^2 = 900\pi$ cm².

Step 3: Volume: $900\pi \times 90 = 81{,}000\pi \approx 254{,}469$ cm³.

Step 4: Litres via the bridge: $\approx 254$ L.

Step 5: Sanity: a bathtub holds ~150–200 L; a hefty rain barrel beating a bathtub by a third feels right ✓. Big-number answers deserve a lived-experience comparison before anyone trusts them.

Example 2 — Reverse engineering: the 1-litre design problem

The brief: design a cylindrical bottle holding exactly 1 L (1000 cm³) with radius 4 cm. Find the height — then critique the design.

Step 1: Set up: $\pi(4)^2 h = 1000$ → $16\pi h = 1000$ .

Step 2: Solve: $h = \frac{1000}{16\pi} \approx 19.9$ cm.

Step 3: Critique like a designer: a 8 cm-wide, 20 cm-tall bottle — plausible water bottle proportions ✓. Re-run with $r = 2$ : $h \approx 79.6$ cm — a metre-ish blowgun; with $r = 10$ : $h \approx 3.2$ cm — a petri dish. Same litre, wildly different objects.

Step 4: The relationship exposed: $h = \frac{1000}{\pi r^2}$ — height falls with the SQUARE of the radius. Doubling width quarters the needed height. The formula isn't just for computing; read as a relationship, it designs.

Example 3 — The composite tank: prism plus cylinder

The tank: a rectangular base section $50 \times 40 \times 30$ cm with a half-cylinder lid along the 50 cm length (diameter 40 cm).

Step 1: Decompose: box + half cylinder.

Step 2: Box: $50 \times 40 \times 30 = 60{,}000$ cm³.

Step 3: Half-cylinder: radius 20, length 50: $\frac{1}{2}\pi(20)^2(50) = \frac{1}{2}(20{,}000\pi) = 10{,}000\pi \approx 31{,}416$ cm³.

Step 4: Total: $\approx 91{,}416$ cm³ ≈ 91.4 L.

Step 5: The decomposition habit, now three units old (composite areas, composite perimeters, composite volumes): complicated shapes are sums of friendly ones. The only new content today was WHICH friendly pieces exist; the strategy is a permanent resident.

MATERIALS

•Cans, boxes, measuring cups

•Circular chips for stacking

•Rulers and calipers

•1 L cube model

•Practice set (PDF)

WATCH FOR

!Diameter used as radius: volumes quadruple. The write-r ritual, again.

!Volume and capacity treated as unrelated (cm³ vs mL panic). One bridge fact, one water demo — connected for good.

!The formula believed to need a "flat-bottomed" orientation: a cylinder lying sideways has the same volume; extrusion doesn't care about gravity.