020i - Volume of Triangular Prisms

Lesson 20 — Volume of Triangular Prisms

Strand: Measurement | Descriptor: AC9M7M02 | Duration: 45 minutes

Learning Intentions

• To understand that the volume of any prism is the cross-sectional area times the length.
• To calculate the volume of a triangular prism using the established formula.

Success Criteria

I can:

1. Identify the triangular cross-section of a prism and distinguish it from the length.
2. Calculate volume using .
3. Apply the two-stage method: find the triangle’s area, then multiply by the length.
4. Find a missing dimension when given the volume.

Warmup

(6 minutes — retrieval, mini whiteboards)

1. Triangle: cm, cm. Find the area.
2. Rectangular prism: cm. Find the volume.
3. A prism has cross-sectional area and length cm. Find its volume.

(Answers: ; ; .)

Bridging question: Question 3 did not say what shape the cross-section was. Did it matter? (No — the formula works for any prism.)

Activities

Activity 1 — Explicit Instruction: the General Prism Formula (12 min)

Definition. A prism is a solid with a constant cross-section along its length — every slice perpendicular to the length is identical.

Physical demonstration. Stack identical paper triangles into a triangular prism, then a deck of cards into a rectangular prism. Each layer contributes the same area; volume is layers accumulated along the length.

The connection to Lesson 19 — make it explicit. Two congruent triangular prisms fit together to form a rectangular prism, exactly as two congruent triangles form a rectangle. So the triangular prism has half the volume of the enclosing rectangular prism.

The two-stage method, modelled slowly. A triangular prism has a cross-section with base cm and height cm, and a length of cm.

Naming warning — the most common source of error. The prism has three different measurements called something like “height”:

• the height of the triangle (perpendicular to the triangle’s base),
• the length of the prism (how far it extends),
• possibly the height it stands at if it is oriented upright.

Insist that students label the diagram with , and before calculating.

We do: cm, cm, cm; m, m, m.

You do: Six triangular prisms, at least two presented as diagrams with a distractor slant edge labelled.

Activity 2 — Working backwards (8 min)

I do. A triangular prism has volume and a cross-sectional area of . Find its length.

A harder case — model the two-stage reversal. A triangular prism has volume , length cm, and a triangular base of cm. Find the height of the triangle.

You do:

1. , . Find .
2. , m. Find .
3. , cm, triangle base cm. Find the triangle’s height.

(Answers: cm; ; , so cm.)

Activity 3 — Inquiry: the Tent and the Ramp (10 min)

Pairs.

Part A. A camping tent is a triangular prism. Its triangular end has base m and height m, and the tent is m long.

1. Find the volume of air inside.
2. A second tent has the same volume but is only m long. If its triangular end has the same base of m, how tall is it?

Part B. A concrete wheelchair ramp is a triangular prism lying on its side: the triangular cross-section has base m (the run) and height m (the rise), and the ramp is m wide.

3. Find the volume of concrete needed.
4. Concrete costs 180$ per cubic metre. Find the cost.

Socratic scaffolding for A2:

Prompt	Purpose
Understand: what is the unknown?	The height of the second tent’s triangular end.
What is the same, and what is different?	Volume and triangle base are the same; length and triangle height differ.
Find the fixed quantity first.	Volume .
Devise a plan	Work backwards: find the new cross-sectional area, then the new height.
Carry it out (area)	.
Carry it out (height)	, so and m.
Looking back	Sensible? The tent is shorter in length, so it must be taller to hold the same air. ✓

Answers: 1. ; 2. m; 3. ; 4. 324$.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Find the volume of a triangular prism with cm, cm, cm.
2. A prism has cross-sectional area and length m. Find its volume.
3. A triangular prism has volume and length cm. Find its cross-sectional area.
4. Reasoning. A student calculates a triangular prism’s volume as and gets . What is the correct answer, and what did they forget?
5. Explain why a triangular prism has exactly half the volume of the rectangular prism that encloses it.

Answers: 1. ; 2. ; 3. ; 4. They forgot to halve — the correct answer is ; 5. Two congruent triangular prisms fit together to form the rectangular prism, so each is half of it.

Common Misconceptions

Misconception	How to pre-empt it
Forgetting to halve, giving .	Require Stage 1 (the triangle’s area) to be written on its own line before multiplying by the length.
Confusing the triangle’s height with the prism’s length.	Insist on labelling the diagram with , and before any calculation.
Using a slant edge of the triangle as its height.	Same trap as Lessons 15–17. Include distractor lengths and require the right-angle mark.
Multiplying all four given numbers when a distractor is present.	Ask “which measurements does the formula actually need?” before starting.
Using for volume.	Link exponent to dimension count; check every final answer’s units.
Assuming any solid with a triangular face is a prism.	A triangular pyramid is not a prism — its cross-section changes along its length.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A triangular prism has volume . If the length is doubled and the cross-sectional area halved, what is the new volume?

Answer

Unchanged.

E2 (AMC Junior style). A rectangular prism cm by cm by cm is cut in half by a diagonal plane through its length, forming two identical triangular prisms. What is the volume of each?

Answer

E3 (Challenge). A triangular prism has volume . Its cross-section is a triangle with base equal to its height. The prism’s length is cm. Find the base of the triangle.

Answer

E4 (Challenge). A trough for animals is a triangular prism, open at the top, with a triangular cross-section of base cm and depth cm, and a length of m. How many litres does it hold when full? (Recall litre.)

Answer

Convert the length: m cm.

Homework

1. Find the volume of each triangular prism: (a) cm, cm, cm (b) m, m, m (c) cm, cm, cm (d) cm, cm, cm.
2. A prism has cross-sectional area and length cm. Find its volume.
3. Find the missing value: (a) , — find (b) , m — find (c) , cm, triangle base cm — find the triangle’s height.
4. A chocolate bar is a triangular prism with cross-sectional base cm, height cm, and length cm. Find its volume.
5. A garden bed is a triangular prism (a wedge) with cross-sectional base m, height m, and length m. Find the volume of soil needed. Soil costs 65$ per cubic metre — find the cost.
6. Reasoning. Explain why works for any prism, whatever the shape of its cross-section.
7. Challenge. A triangular prism and a rectangular prism have the same volume. The rectangular prism is cm. The triangular prism has length cm and a cross-section whose base is cm. Find the triangle’s height.

Answers: Q1 — (a) (b) (c) (d) . Q2 — . Q3 — (a) cm (b) (c) , so cm. Q4 — . Q5 — ; cost 62.40V = 120\ \text{cm}^3A = 12\ \text{cm}^2\tfrac12 \times 8 \times h = 12h = 3$ cm.