026i - Volume of Right Prisms

Lesson 26 — Explicit Instruction: Volume of Right Prisms Using Cross-Sectional Area

Strand: Measurement | Descriptor: AC9M8M02 | Duration: 45 minutes

Learning Intentions

• To understand that a right prism’s volume equals the area of its uniform cross-section multiplied by its length.
• To calculate the volume of rectangular, triangular and trapezoidal right prisms using .

Success Criteria

I can:

1. Identify the uniform cross-section of a right prism and state its shape.
2. Calculate the area of a rectangular, triangular or trapezoidal cross-section.
3. Apply to find the volume of a right prism.
4. Explain why the "" in this formula is the prism’s length, not necessarily a vertical distance.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

Recall these area formulas:

1. Area of a rectangle with sides cm and cm.
2. Area of a triangle with base cm and height cm.
3. Area of a trapezium with parallel sides cm and cm, and height cm.
4. If you stacked identical flat shapes, each of area , directly on top of one another to make a solid block, what would the total volume be?

Answers: 1. ; 2. ; 3. ; 4. .

The hook: Question 4 is exactly how today’s formula works — a right prism is just a stack of identical cross-sections, and its volume is the area of one layer multiplied by how many layers (the length) it has.

Activities

Activity 1 — Explicit Instruction: the Cross-section and the Volume Formula (12 min)

Definition. A right prism is a 3D solid with two congruent, parallel end faces (the cross-section) of identical shape and size along its entire length, connected by rectangular side faces perpendicular to the cross-section.

The formula:

where is the area of the constant 2D shape, and is the length of the prism (how far that shape is “extruded”).

I do — rectangular prism. A rectangular prism has cross-section and length cm.

Note explicitly: this is the same as the familiar — that formula is just this one applied to a rectangular cross-section.

I do — triangular prism. A triangular prism has a triangular cross-section with base cm and height cm, and the prism itself is cm long.

Critical distinction to model explicitly: the ” cm” is the height of the triangle (part of the area calculation); the ” cm” is the length of the prism (how far the triangle is extruded). These are two completely different measurements, both loosely called “height” in everyday speech — the context tells you which is which.

We do — trapezoidal prism. Cross-section is a trapezium with parallel sides cm and cm, trapezium height cm; prism length cm.

You do:

1. Rectangular prism, cross-section , length cm.
2. Triangular prism, triangle base cm, triangle height cm, prism length cm.
3. Trapezoidal prism, parallel sides cm and cm, trapezium height cm, prism length cm.

(Answers: 1. , ; 2. , ; 3. , .)

Activity 2 — Guided Practice: Identifying the Cross-section Correctly (10 min)

Pairs. For each prism, first state the shape of the cross-section, then calculate its area, then the volume.

1. A prism standing on a triangular base, base cm, triangle height cm, prism length cm.
2. A prism lying on its side, so its rectangular cross-section is , and the prism’s length is cm.
3. A trapezoidal prism used as a garden edging beam: parallel sides cm and cm, trapezium height cm, beam length cm.
4. A triangular prism with a right-angled triangular cross-section, legs cm and cm, prism length cm.

Discussion point for Q2 and Q4: in Q2, the prism is drawn lying down, but the formula still works exactly the same way — the “length” is simply whichever dimension is constant along the prism’s axis, regardless of how the solid is oriented on the page. In Q4, remind students the two legs of a right-angled triangle are its base and height for area purposes.

Answers: 1. , ; 2. , ; 3. , ; 4. , .

Activity 3 — Inquiry: Does the Formula Really Work for Any Right Prism? (12 min)

Pairs, then whole-class share.

A rectangular prism measures . It is cut by a single flat diagonal cut, along its length, into two identical triangular prisms.

1. Find the volume of the original rectangular prism.
2. Predict the volume of each triangular half.
3. Use to verify your prediction.

Socratic scaffolding:

Prompt	Purpose
Understand: what shape results from cutting a rectangular prism along a diagonal plane?	Two congruent triangular prisms, each with the same length as the original.
What is the cross-sectional area of the original rectangular prism?	.
What fraction of that rectangle is each triangular cross-section?	Exactly half — a diagonal splits a rectangle into two congruent right-angled triangles.
So what is the area of each triangular cross-section?	.
Devise a plan using	Multiply the triangular cross-section’s area () by the prism’s length ( cm).
Carry it out	.
Looking back — check against the whole block	Original volume . Two triangular prisms should together equal this: ✓
Looking back — does this generalise?	Yes — because area itself splits proportionally when a shape is halved, the volume formula automatically respects that split, for any right prism, not just rectangular ones.

Answers: 1. . 2. each. 3. Confirmed above.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. State, in your own words, what "" represents in the formula .
2. A rectangular prism has cross-section and length cm. Find its volume.
3. A triangular prism has a triangular cross-section with base cm and height cm, and a length of cm. Find its volume.
4. A trapezoidal prism has parallel sides cm and cm, trapezium height cm, and prism length cm. Find its volume.
5. Reasoning. A student says “the height of a prism is always the vertical measurement in the picture.” Explain, using Activity 2 Question 2, why this is not always true.

Answers: 1. The length of the prism — how far the constant cross-sectional shape extends, which is not necessarily a vertical measurement; 2. , ; 3. , ; 4. , ; 5. In Activity 2 Q2 the prism was drawn lying on its side, so its length ran horizontally across the page, not vertically — “height” in the formula means the direction of extrusion, whichever way the solid happens to be drawn.

Common Misconceptions

Misconception	How to pre-empt it
Confusing the prism’s length ( in the formula) with the height of the 2D cross-sectional shape.	Model both measurements with different colours in every diagram, as in Activity 1’s triangular prism example.
Believing only works for rectangular prisms and is unrelated to other shapes.	Explicitly show that is just with a rectangular cross-section.
Forgetting to halve the base height product when the cross-section is a triangle.	Insist the cross-sectional area is calculated and written as a separate labelled line before multiplying by length.
Multiplying all given lengths together regardless of shape.	Require students to first name the cross-sectional shape and its correct area formula before any multiplication.
Writing the final answer without cubic units, or using squared units for volume.	Model units at every line: for the cross-section, for the volume.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A triangular prism has volume and length cm. If the triangular cross-section has a base of cm, find its height.

Answer

E2 (AMC Junior style). Two rectangular prisms have the same volume. The first has cross-section and length cm. The second has the same cross-section but an unknown length . Explain what this tells you about , and find its value.

Answer

Since both prisms have identical cross-sections and equal volumes, they must have equal lengths: . (A nice check that volume depends on cross-section and length together, not on shape alone.)

E3 (Challenge). A triangular prism and a rectangular prism have equal volumes and equal lengths. The rectangular prism’s cross-section is . If the triangular prism’s cross-section has a base of cm, find its height.

Answer

Equal volumes and equal lengths mean equal cross-sectional areas: . Then .

E4 (Investigation). A right prism’s cross-section is a regular hexagon made of identical equilateral triangles, each of area . The prism is cm long. Find its volume.

Answer

This shows the formula works for any uniform cross-section, however many sides it has, as long as the total cross-sectional area is known.

Homework

1. Find the volume of a rectangular prism with cross-section and length cm.
2. Find the volume of a triangular prism with triangular base cm, triangular height cm, and prism length cm.
3. Find the volume of a trapezoidal prism with parallel sides cm and cm, trapezium height cm, and prism length cm.
4. A triangular prism has a right-angled triangular cross-section with legs cm and cm, and a length of cm. Find its volume.
5. Reasoning. Explain, using the formula , why doubling only the prism’s length doubles its volume, but doubling only one side of a triangular cross-section does not double the volume.
6. Challenge. A right prism has a triangular cross-section of area and volume . Find its length. If the triangle’s base is cm, find the triangle’s height.

Answers: Q1 — , . Q2 — , . Q3 — , . Q4 — , . Q5 — doubling the length multiplies the volume by exactly , since appears once, linearly, in the formula; doubling one side of a triangle changes the cross-sectional area, but only by the same factor as that one side (since the other side and the stay fixed), so the area — and hence the volume — is scaled by , not , and the shape of the triangle also changes, unlike a uniform scale of both legs. Q6 — length cm; height cm.