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Lesson 10 ~25 min Unit 3 · Measurement & Geometry +85 XP

Volume of Prisms

One rule covers every prism: $V = A_\text{base} \times h$ — find the cross-section area, then multiply by the length.

Today's hook: A swimming pool is 3 m wide, 1.5 m deep and 12 m long. How many litres does it hold? (1 m³ = 1000 L)
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Think First
warm-up

If you stack 1-cm cubes to fill a 3×4×2 box, how many cubes fit? Without counting every cube, how could you work it out faster?

Record your answer in your workbook.
1
The Big Idea
+5 XP

Volume measures the amount of 3D space a solid occupies. For any prism, the formula is: $V = A_\text{base} \times h$, where $A_\text{base}$ is the area of the cross-section and $h$ is the height (length) of the prism. The cross-section must be the face that stays constant as you move along the prism.

A h V = Aₛàₛá × h
$$V = A_\text{base} \times h$$
Identify the cross-section first
The base is the shape that is constant as you slice along the prism. For a triangular prism it is the triangle, not one of the rectangular sides.
Cubic units
Volume is measured in cubic units: cm³, m³, mm³. Not square units (that is surface area).
Capacity link
1 cm³ = 1 mL. 1 000 cm³ = 1 L. 1 m³ = 1 000 L. Always check which unit you need for the answer.
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What You'll Master
objectives

Know

  • $V = A_\text{base} \times h$ for any prism
  • Rectangular: $V = lwh$
  • Triangular: $V = \frac{1}{2}bh \times l$
  • 1 cm³ = 1 mL; 1 000 cm³ = 1 L; 1 m³ = 1 000 L

Understand

  • Why the same formula works for every prism (constant cross-section)
  • How to identify the correct base for non-rectangular prisms
  • The difference between volume (cm³) and capacity (L, mL)

Can Do

  • Find volume of rectangular, triangular, and composite prisms
  • Convert volume answers to capacity (litres, millilitres)
  • Solve real-world filling and pool problems
3
Words You Need
vocabulary
VolumeThe amount of 3D space a solid occupies. Measured in cubic units (cm³, m³, mm³).
PrismA solid with a constant cross-section along its length. The two end faces are congruent and parallel.
Cross-sectionThe shape you get when you cut through the prism at right angles to its length. This is the "base" in $V = A_\text{base} \times h$.
CapacityThe amount of liquid a container can hold. 1 cm³ = 1 mL; 1 000 cm³ = 1 L; 1 m³ = 1 000 L.
Rectangular prismA box shape. Cross-section is a rectangle. $V = l \times w \times h$.
Triangular prismCross-section is a triangle. $V = \frac{1}{2}bh_\triangle \times l$, where $l$ is the prism length.
4
Spot the Trap
heads-up

Classic error: confusing the triangle's height with the prism's length in a triangular prism.

Wrong (used prism length as triangle height)

$V = \frac{1}{2} \times 8 \times 12 \times 5 = 240$ cm³ ✗  (used $l = 12$ as $h$)

Correct

$A_\triangle = \frac{1}{2} \times 8 \times 5 = 20$ cm² then $V = 20 \times 12 = 240$ cm³ ✓

Rule: Find $A_\text{base}$ first using the triangle's own base and height, then multiply by the prism's length.

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Volume of Rectangular Prisms
+5 XP

For a rectangular prism (cuboid), the cross-section is a rectangle with area $A = l \times w$. Multiplying by the height $h$ gives:

$V = l \times w \times h$

This is also written $V = lwh$. Any order of multiplication works: $l \times w \times h = w \times l \times h$.

$l = 5,\ w = 4,\ h = 3$ $A_\text{base} = 5 \times 4 = 20$ cm² $V = 20 \times 3 = 60$ cm³
$$V = lwh$$
Any order
$5 \times 4 \times 3 = 4 \times 3 \times 5 = 60$. Multiplication is commutative, so choose the easiest order.
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Volume of Triangular Prisms
+5 XP

The cross-section is a triangle. Find its area first: $A_\triangle = \frac{1}{2} \times b \times h_\triangle$, where $b$ and $h_\triangle$ are the triangle's base and perpendicular height. Then multiply by the prism length $l$:

$V = \frac{1}{2} \times b \times h_\triangle \times l$

$b = 8,\ h_\triangle = 5,\ l = 12$ $A_\triangle = \frac{1}{2} \times 8 \times 5 = 20$ cm² $V = 20 \times 12 = 240$ cm³
$$V = \tfrac{1}{2}\,b\,h_\triangle \times l$$
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Volume of Composite and Trapezoidal Prisms
+5 XP

For an L-shaped cross-section, split it into two rectangles, find both areas, add them, then multiply by the prism length.

For a trapezium cross-section: $A = \frac{1}{2}(a+b)h$ where $a$ and $b$ are the parallel sides. Then $V = A \times l$.

R1 R2 + $A = A_{R1} + A_{R2}$ $V = A \times l$
$A_\text{L-shape} = A_{R1} + A_{R2}$  then  $V = A \times l$
8
Capacity Conversions
+5 XP

Volume and capacity are related by:

  • $1\ \text{cm}^3 = 1\ \text{mL}$
  • $1\ 000\ \text{cm}^3 = 1\ \text{L}$
  • $1\ \text{m}^3 = 1\ 000\ \text{L}$
  • $1\ \text{m}^3 = 1\ 000\ 000\ \text{mL}$

To convert cm³ to mL: same number. To convert cm³ to L: divide by 1 000.

cm³ $\xrightarrow{\div 1\,000}$ L $\xrightarrow{\times 1\,000}$ mL m³ $\xrightarrow{\times 1\,000}$ L e.g. $54\ \text{m}^3 = 54 \times 1\,000 = 54\,000\ \text{L}$
$1\ \text{cm}^3 = 1\ \text{mL}$    $1\ 000\ \text{cm}^3 = 1\ \text{L}$    $1\ \text{m}^3 = 1\ 000\ \text{L}$
Check units first
If the question gives dimensions in cm but asks for litres, compute in cm³ then divide by 1 000. If in m then multiply m³ by 1 000.

Watch Me Solve It · 3 examples

WE 1 — Volume of a Rectangular Prism
+10 XP
Q1
PROBLEM
Find the volume of a rectangular box: $l = 5$ cm, $w = 4$ cm, $h = 3$ cm.
  1. 1
    Write the formula
    $$V = lwh$$
  2. 2
    Substitute values
    $V = 5 \times 4 \times 3$
  3. 3
    Calculate
    $V = 60$ cm³
Answer$V = 60$ cm³
WE 2 — Volume of a Triangular Prism
+10 XP
Q2
PROBLEM
A triangular prism has triangle base $b = 8$ cm, triangle height $h = 5$ cm, and prism length $l = 12$ cm. Find its volume.
  1. 1
    Find the triangle cross-section area
    $A_\triangle = \frac{1}{2} \times b \times h = \frac{1}{2} \times 8 \times 5 = 20$ cm²
  2. 2
    Multiply by prism length
    $V = A_\triangle \times l = 20 \times 12 = 240$ cm³
Answer$V = 240$ cm³
WE 3 — Pool Volume and Capacity
+10 XP
Q3
PROBLEM
A swimming pool is 3 m wide, 1.5 m deep and 12 m long. Find its volume in m³ and its capacity in litres.
  1. 1
    Write the formula
    $$V = lwh$$
  2. 2
    Substitute values
    $V = 12 \times 3 \times 1.5 = 54$ m³
  3. 3
    Convert to litres
    $54\ \text{m}^3 \times 1\,000 = 54\,000\ \text{L}$
Answer$V = 54\ \text{m}^3 = 54\,000\ \text{L}$
9
Common Pitfalls
heads-up
Confusing Triangle Height With Prism Length
Mistake: $V = \frac{1}{2} \times b \times l \times l$ (using length twice).
Fix: Label carefully. $b$ and $h_\triangle$ belong to the triangle; $l$ is the prism's length. Compute $A_\triangle$ first.
Giving Area Units Instead of Volume Units
Mistake: Writing the answer as cm² instead of cm³.
Fix: Volume is always cm³, m³, etc. If your answer has square units, you forgot to multiply by the third dimension.
Wrong Conversion Factor for Litres
Mistake: Multiplying m³ by 100 instead of 1 000.
Fix: 1 m³ = 1 000 L (not 100). Remember: $1\ \text{m} = 100\ \text{cm}$, so $1\ \text{m}^3 = 100^3\ \text{cm}^3 = 1\,000\,000\ \text{cm}^3 = 1\,000\ \text{L}$.
Copy Into Your Books
Any prism: $V = A_\text{base} \times h$
Rectangular prism: $V = lwh$
Triangular prism: $V = \frac{1}{2}bh_\triangle \times l$ — find triangle area first
Capacity: 1 cm³ = 1 mL   1 000 cm³ = 1 L   1 m³ = 1 000 L

How are you completing this lesson?

Brain Trainer · 4 problems

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Brain Trainer · Volume of Prisms
4 problems

Set a timer for 4 minutes. Show all working.

  1. 1 Find V of a cube with side 6 cm.

    $V = 6 \times 6 \times 6 = 216$ cm³

  2. 2 Triangular prism: $b = 10$ cm, $h_\triangle = 6$ cm, $l = 8$ cm.

    $A = \frac{1}{2}(10)(6) = 30$ cm². $V = 30 \times 8 = 240$ cm³

  3. 3 Box 20 cm × 15 cm × 10 cm. Find volume in cm³ then in litres.

    $V = 20 \times 15 \times 10 = 3\,000$ cm³ $= 3$ L

  4. 4 Hook pool: 3 m × 1.5 m × 12 m. Find volume and capacity in litres.

    $V = 3 \times 1.5 \times 12 = 54$ m³ $= 54\,000$ L
Complete in your workbook.

Quick Check · 5 questions

1
Which formula gives the volume of any prism?
+10 XP
2
Volume of a box: $l = 4$ cm, $w = 5$ cm, $h = 6$ cm.
+10 XP
3
Triangular prism: $b = 8$ cm, $h_\triangle = 5$ cm, $l = 12$ cm.
+10 XP
4
A box 15 cm × 10 cm × 10 cm. What is its capacity in mL?
+10 XP
5
A pool is 4 m long, 2 m wide, 3 m deep. How many litres does it hold?
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
ApplyMedium3 MARKS

Q6. A rectangular storage box is 8 cm × 5 cm × 6 cm. Find its volume in cm³ and its capacity in millilitres.

Show full working in your book.
ApplyMedium3 MARKS

Q7. A triangular prism has a right-angled triangular cross-section with legs 9 cm and 12 cm. The prism is 20 cm long. Find its volume.

Show full working in your book.
ReasonHard3 MARKS

Q8. An L-shaped cross-section prism is 15 cm long. The L-shape can be split into two rectangles: one 8 cm × 4 cm and one 6 cm × 3 cm. Find the prism's volume.

Show full working in your book.
Comprehensive Answers

MC: 1-B, 2-C, 3-A, 4-B, 5-D

Q6: $V = 8 \times 5 \times 6 = 240$ cm³ $= 240$ mL.

Q7: $A_\triangle = \frac{1}{2} \times 9 \times 12 = 54$ cm². $V = 54 \times 20 = 1\,080$ cm³.

Q8: $A_\text{total} = 32 + 18 = 50$ cm². $V = 50 \times 15 = 750$ cm³.

Stretch Challenge · +25 XP

Fish Tank Problem

A rectangular fish tank is 80 cm long, 40 cm wide and 50 cm tall. Water is filled to a depth of 40 cm.

(a) How many litres of water are in the tank?

(b) 8 litres evaporate. By how many centimetres does the water level drop?

Reveal solution

(a) $V = 80 \times 40 \times 40 = 128\,000$ cm³ $= 128$ L.

(b) $8$ L $= 8\,000$ cm³. Base area $= 80 \times 40 = 3\,200$ cm². Drop $= 8\,000 \div 3\,200 = 2.5$ cm.

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Quick Review

Any prism

$V = A_\text{base} \times h$

Rectangular prism

$V = lwh$

Triangular prism

$V = \frac{1}{2}bh_\triangle \times l$

Composite prism

Split cross-section, add areas, then $\times h$

1 cm³ = 1 mL

1 000 cm³ = 1 L   1 m³ = 1 000 L

Key pitfall

Triangular prism: use triangle's $h$, not prism length, for $A_\triangle$

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Triangular prism: $b = 6$ cm, $h_\triangle = 4$ cm, $l = 10$ cm. Find volume in under 60 seconds for double coins!
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