Mathematics • Year 9 • Unit 4 • Lesson 6
Volume of Prisms and Cylinders
Build fluency with $V = A_{\text{cross}} \times h$ for prisms and $V = \pi r^2 h$ for cylinders — one step at a time, from a fully worked example through guided practice to independent problems.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. A cylinder has radius 5 cm and height 10 cm. Find its volume (a) in terms of $\pi$, and (b) to 1 decimal place.
Step 1 — Spot the rule.
A cylinder has a circular cross-section, so $V = \pi r^2 h$.
Reason: the cross-section is a circle of area $\pi r^2$, and the cylinder is just that circle "stacked" up to height $h$.
Step 2 — Substitute the numbers.
$V = \pi \times (5)^2 \times 10$
Reason: $r = 5$, $h = 10$. Square the radius FIRST, before multiplying by $\pi$ and $h$.
Step 3 — Square the radius.
$(5)^2 = 25$, so $V = \pi \times 25 \times 10 = 250\pi$ cm³
Reason: this is the exact answer. Leave it as $250\pi$ when "in terms of $\pi$" is asked.
Step 4 — Get a decimal answer.
$250\pi \approx 250 \times 3.14159 \approx 785.4$ cm³
Reason: use the $\pi$ key on your calculator. Round only at the very end to the asked number of decimal places.
Answer: (a) $\mathbf{250\pi}$ cm³, (b) $\mathbf{\approx 785.4}$ cm³.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Find the volume of a triangular prism with base 6 cm, perpendicular height 4 cm, and length 10 cm.
Step 1 — Spot the rule: a prism's volume is $V = A_{\text{cross}} \times \_\_\_\_\_\_\_\_$. The cross-section is a __________________ .
Step 2 — Area of the cross-section (triangle):
$A = \dfrac{1}{2} \times \text{base} \times \text{height} = \dfrac{1}{2} \times \_\_\_ \times \_\_\_ = \_\_\_\_$ cm²
Step 3 — Multiply by the length of the prism:
$V = A \times \text{length} = \_\_\_\_ \times 10$
Step 4 — State the answer with units:
$V = \_\_\_\_\_\_\_\_$ cm³
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single formula). The middle two are standard (combine units or pick the right formula). The last two are extension (capacity, unit conversion).
Foundation — single formula
3.1 Find the volume of a rectangular prism 7 cm × 4 cm × 3 cm. 1 mark
3.2 Find the volume of a cube of side 8 cm. 1 mark
3.3 Find the volume of a cylinder with radius 3 cm and height 8 cm. Leave your answer in terms of $\pi$. 1 mark
3.4 Find the volume of a triangular prism with base 5 cm, perpendicular height 6 cm, and length 12 cm. 1 mark
Standard — pick the right formula
3.5 A cylinder has diameter 10 cm and height 4 cm. Find its volume in terms of $\pi$. 2 marks
3.6 A rectangular fish tank measures 80 cm × 40 cm × 30 cm. Find its volume in cm³ and convert to litres. (1000 cm³ = 1 L.) 2 marks
Extension — push your thinking
3.7 A cylindrical water tank has radius 1.5 m and height 2 m. How many litres does it hold? (1 m³ = 1000 L.) Give your answer to the nearest litre. 3 marks
3.8 A rectangular prism of volume $216$ cm³ has length 8 cm and width 3 cm. Find its height. Explain which formula you rearranged. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (triangular prism 6 × 4 × 10)
Step 1: $V = A_{\text{cross}} \times \mathbf{\text{length}}$. The cross-section is a triangle.
Step 2: $A = \tfrac{1}{2} \times \mathbf{6} \times \mathbf{4} = \mathbf{12}$ cm².
Step 3: $V = \mathbf{12} \times 10$.
Step 4: $V = \mathbf{120}$ cm³.
3.1 — Rectangular prism 7 × 4 × 3
$V = l \times w \times h = 7 \times 4 \times 3 = \mathbf{84}$ cm³.
3.2 — Cube of side 8
$V = 8^3 = 8 \times 8 \times 8 = \mathbf{512}$ cm³.
3.3 — Cylinder $r = 3$, $h = 8$
$V = \pi r^2 h = \pi \times 9 \times 8 = \mathbf{72\pi}$ cm³ ($\approx 226.2$ cm³).
3.4 — Triangular prism (base 5, height 6, length 12)
$A = \tfrac{1}{2} \times 5 \times 6 = 15$ cm². $V = 15 \times 12 = \mathbf{180}$ cm³.
3.5 — Cylinder $d = 10$, $h = 4$
Radius $r = 10 \div 2 = 5$ cm. $V = \pi \times 5^2 \times 4 = \mathbf{100\pi}$ cm³ ($\approx 314.2$ cm³).
Common slip: using $d = 10$ in $\pi r^2$ gives $400\pi$ — four times too big. Always halve to find $r$ first.
3.6 — Fish tank 80 × 40 × 30 cm
$V = 80 \times 40 \times 30 = 96\,000$ cm³.
Convert: $96\,000 \div 1000 = \mathbf{96}$ L.
3.7 — Water tank $r = 1.5$ m, $h = 2$ m
$V = \pi \times (1.5)^2 \times 2 = \pi \times 2.25 \times 2 = 4.5\pi$ m³ $\approx 14.137$ m³.
Convert m³ → L: $14.137 \times 1000 \approx \mathbf{14\,137}$ L.
3.8 — Find the missing height
Rearrange $V = lwh$ to $h = \dfrac{V}{lw} = \dfrac{216}{8 \times 3} = \dfrac{216}{24} = \mathbf{9}$ cm.
I rearranged the rectangular prism formula because volume, length and width are known and height is the unknown.