LESSON PLAN

Volume of Triangular Prisms

Apothem Team

Grade 6 · Measurement

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Hold up a cereal box and a Toblerone-style triangular box: "Which holds more, and how would you PROVE it without pouring?" Collect proposals — most reach for filling with something countable.

Anchor the definition on that instinct: volume is a count of unit cubes that fit inside. Last year rectangular boxes surrendered to $\ell \times w \times h$ ; today the triangular prism joins them, and the method (layers) is the star, not the formula.

Explore

Layer lab: pairs build a rectangular prism from centimetre cubes ( $4 \times 3$ base, 5 layers) and re-derive the formula as (cubes in one layer) × (number of layers) = base area × height.

Then the handoff: a triangular prism is HALF a rectangular box — demonstrate by slicing a rectangular clay prism diagonally through its base. So its volume is (triangle base area) × height. Pairs compute one triangular prism's volume from measurements and check by water displacement or rice-filling into a measuring cup if time allows.

Formalize

Formalize the one formula that covers every prism: volume = area of the base (the congruent cross-section) times the height (how far that cross-section is extruded):

$V_{\text{prism}} = A_{\text{base}} \times h \qquad A_{\text{triangle}} = \tfrac{1}{2} b h_\triangle \;\Rightarrow\; V = \tfrac{1}{2} b h_\triangle \times H$

Units discipline: volume comes in CUBIC units (cm³, m³) — cubes, not tiles, not lengths. And the two heights must never blur: the triangle's own height $h_\triangle$ lives inside the base's area; the prism's height $H$ is the extrusion length. Label both on every diagram.

Practice

Practice: two rectangular prisms (one with decimal edges), three triangular prisms in varied orientations (including one lying on a rectangular face so the "height" runs sideways), and one reverse problem: given $V$ and the base area, find the height.

Exit ticket: a tent has a triangular cross-section 2 m wide and 1.5 m tall, and is 3 m long — find its volume. ( $\frac{1}{2}(2)(1.5)\times3 = 4.5$ m³.)

Exit ticket

Exit ticket: a tent has a triangular cross-section 2 m wide and 1.5 m tall, and is 3 m long — find its volume. ( $\frac{1}{2}(2)(1.5)\times3 = 4.5$ m³.)

TIP Orientation is the enemy: rotate every prism model until students can point to the two congruent bases regardless of which face it rests on. The base is the repeated cross-section, not the face on the table.

WORKED EXAMPLES

Example 1 — The wedge of cheese: a triangular prism on its side

The cheese: a triangular prism — triangle base 8 cm, triangle height 6 cm, prism length 10 cm — photographed lying on a rectangular face, so nothing points "up."

Step 1: Find the true bases: the two triangle faces at the ends. (Rotate the picture mentally until they're top and bottom if it helps.)

Step 2: Base area: $A = \frac{1}{2}(8)(6) = 24$ cm².

Step 3: Extrude through the prism's length: $V = 24 \times 10 = 240$ cm³.

Step 4: Reasonableness: the bounding rectangular box would be $8 \times 6 \times 10 = 480$ cm³, and the wedge is half of it: $240$ ✓ — the slicing demonstration, now working as a checking device.

Example 2 — Reverse engineering: how tall is the aquarium?

The aquarium: rectangular base 60 cm by 25 cm; it holds 45,000 cm³ (45 L) when full. How tall is it?

Step 1: Write the formula and fill what's known: $V = A_{\text{base}} \times h$ → $45{,}000 = (60 \times 25) \times h$ .

Step 2: Base area: $1{,}500$ cm².

Step 3: Solve for the height: $h = 45{,}000 \div 1{,}500 = 30$ cm.

Step 4: Check the litre conversion while it's warm: $1$ L $= 1{,}000$ cm³, so 45 L ↔ 45,000 cm³ ✓.

Step 5: Name the move: formulas run backwards via division — one formula is really three ( $V = Ah$ , $A = V/h$ , $h = V/A$ ), the fact-family idea all over again, now wearing geometry.

Example 3 — Same volume, different shape: the chocolate-bar redesign

The pitch: a company sells a rectangular bar $10 \times 4 \times 3$ cm and wants a triangular-prism version with the SAME 120 cm³ of chocolate, keeping the length 10 cm. Design the triangle.

Step 1: Required base area: $V = A \times 10 = 120$ → $A = 12$ cm² for the triangular cross-section.

Step 2: Choose triangle dimensions with $\frac{1}{2}bh = 12$ → $bh = 24$ : options include $b=8, h=3$ ; $b=6, h=4$ ; $b=4, h=6$ (tall and dramatic on the shelf).

Step 3: Check one: $\frac{1}{2}(6)(4) \times 10 = 120$ ✓.

Step 4: The design discussion that makes it real: same chocolate, but the tall triangle LOOKS bigger in packaging, and its box uses different amounts of cardboard (surface area — next unit's cliffhanger). Volume fixed, everything else negotiable: that's engineering.

MATERIALS

•Centimetre cubes

•Triangular and rectangular prism models

•Clay and dental floss for the diagonal slice

•Rice and measuring cup

•Practice set (PDF)

WATCH FOR

!The prism's height confused with the triangle's height — the two-heights labelling ritual targets this directly.

!Base = "the face it sits on." The base pair is the congruent cross-section faces, wherever they point.

!Volume reported in square units. Cubes fill; tiles cover — the unit names the dimension.