Volume of solids

Unit 4 Volume of solids

Key unit competence: Select and use appropriate formulae to find the volume of solids.

Key ideas

• Solids have faces, edges and vertices. For polyhedra, Euler's formula: $F+V-E=2$ (a cube: $6+8-12=2$).
• A prism has a uniform cross-section: volume = area of cross-section × length. A cuboid: $V=lwh$; a cube: $V=a^3$.
• A cylinder is a circular prism: $V=\pi r^{2}h$.
• A cone holds one third of the cylinder with the same base and height: $V=\frac{1}{3}\pi r^{2}h$.
• A pyramid holds one third of the prism with the same base and height: $V=\frac{1}{3}\times \text{base area}\times h$.
• A sphere of radius $r$: $V=\frac{4}{3}\pi r^3$. A hemisphere is half of it.
• Units: $1{\text{ m}}^3=1000\text{ litres}$, $1\text{ litre}=1000{\text{ cm}}^3$.

Formulas to know

Common misconceptions

• "Use the diameter in $\pi r^{2}h$." No: use the radius, which is half of the diameter.
• "Volume of a cone $=\pi r^{2}h$." No: it is one third of that.
• "1 m³ = 100 litres." No: $1{\text{ m}}^3=1000$ litres.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Verify Euler's formula for a triangular prism.

   Show answer

   $F=5$, $V=6$, $E=9$: $5+6-9=2$. ✓

   Common mistake: Counting the edges of each face separately (counting shared edges twice).

2. A cylinder of height 12 cm has volume 1 356.48 cm³ ($\pi \approx 3.14$). Find its radius.

   Show answer

   $r^2=\frac{1356.48}{3.14\times 12}=36$, $r=6$ cm.

   Common mistake: Forgetting to take the square root.

3. Find the volume of a cylinder of radius 7 cm and height 20 cm ($\pi =\frac{22}{7}$). Give the answer in litres.

   Show answer

   $\frac{22}{7}\times 49\times 20=3080$ cm³ $=3.08$ L.

   Common mistake: Dividing by 100 instead of 1000 to get litres.

4. An ice-cream cone (radius 3 cm, height 10 cm) is filled and topped with a hemisphere of ice cream of the same radius. Find the total volume of ice cream in terms of $\pi$.

   Show answer

   Cone $\frac{1}{3}\pi \times 9\times 10=30\pi$; hemisphere $\frac{2}{3}\pi \times 27=18\pi$; total $48\pi$ cm³.

   Common mistake: Using a full sphere on top.

5. A cone has radius 6 cm and slant height 10 cm. Find its height and volume ($\pi \approx 3.14$).

   Show answer

   $h=\sqrt{10^2-6^2}=8$ cm; $V=\frac{1}{3}\times 3.14\times 36\times 8=301.44$ cm³.

   Common mistake: Using the slant height as the height.

6. A swimming pool is 25 m long, 10 m wide and 2 m deep. How many litres of water does it hold when full?

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   $500$ m³ $=500000$ litres.

   Common mistake: Leaving the answer in m³ when litres were asked.

7. A pyramid has a rectangular base 8 cm by 6 cm and height 9 cm. Find its volume.

   Show answer

   $\frac{1}{3}\times 48\times 9=144$ cm³.

   Common mistake: Forgetting the $\frac{1}{3}$.

8. A spherical ball has diameter 21 cm. Find its volume ($\pi =\frac{22}{7}$).

   Show answer

   $r=10.5$; $\frac{4}{3}\times \frac{22}{7}\times 10.5^3=4851$ cm³.

   Common mistake: Using 21 as the radius.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. A metal cube of side 6 cm is melted and recast as a cylinder of radius 3 cm. Find the height of the cylinder (1 d.p.).	a. 7.6 cm
   2. A prism has a cross-section of area 15 cm² and length 8 cm. Find its volume.	b. $180\pi$ cm³
   3. A pipe is 20 cm long with outer radius 5 cm and inner radius 4 cm. Find the volume of material.	c. 120 cm³
   4. A water tank holds 2.5 m³. How many litres is this?	d. 2 500 L

   Show answer

   1 → a (7.6 cm) · 2 → c (120 cm³) · 3 → b ($180\pi$ cm³) · 4 → d (2 500 L)

   • Step 1. Volume of metal $=6^3=216$ cm³.
     Step 2. $\pi \times 3^2\times h=216$, so $h=\frac{216}{9\pi }=\frac{24}{\pi }\approx 7.6$ cm.
   • $V=A\times l=15\times 8$.
   • Step 1. Cross-section: $\pi (5^2-4^2)=9\pi$ cm².
     Step 2. $V=9\pi \times 20=180\pi$ cm³.
   • $2.5\times 1000=2500$ litres.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   9 times larger  ·  8  ·  15 cm  ·  27 times larger
   1. If the side of a cube is doubled, its volume is multiplied by .
   2. Radius of a sphere is tripled. Its volume becomes .
   3. A cone and a cylinder have the same radius and the same volume. If the cylinder's height is 5 cm, the cone's height is .
   Show answer

   1: 8 · 2: 27 times larger · 3: 15 cm

   • $(2a{)}^3=8a^3$.
   • Volume depends on $r^3$: $3^3=27$.
   • $\frac{1}{3}\pi r^{2}h=\pi r^2\times 5\Rightarrow h=15$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. A cylinder of height 4 cm has volume 154 cm³. Find its radius ($\pi =\frac{22}{7}$).	a. 60 cm³
   2. A cylinder has diameter 10 cm and height 7 cm. Use $\pi =\frac{22}{7}$. Find its volume.	b. 3.5 cm
   3. A pyramid has a square base of side 6 cm and height 5 cm. Find its volume.	c. 550 cm³
   4. A sphere has volume $36\pi$ cm³. Find its radius.	d. 3 cm

   Show answer

   1 → b (3.5 cm) · 2 → c (550 cm³) · 3 → a (60 cm³) · 4 → d (3 cm)

   • Step 1. $\pi r^{2}h=154$: $\frac{22}{7}\times r^2\times 4=154$, so $r^2=\frac{154\times 7}{88}=12.25$.
     Step 2. $r=\sqrt{12.25}=3.5$ cm.
   • $r=5$: $\frac{22}{7}\times 25\times 7=550$.
   • $\frac{1}{3}\times 36\times 5=60$.
   • Step 1. $\frac{4}{3}\pi r^3=36\pi$, so $r^3=27$.
     Step 2. $r=3$ cm.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $\frac{2}{3}\pi r^3$  ·  About 94 minutes  ·  $36\pi$ cm³  ·  $108\pi$ cm³
   1. The volume of a hemisphere of radius $r$ is .
   2. Water flows into a cylindrical tank of radius 1 m at 50 litres per minute. How long to fill it to a depth of 1.5 m? ($\pi \approx 3.14$) →
   3. Find the volume of a sphere of radius 3 cm, in terms of $\pi$. →
   Show answer

   1: $\frac{2}{3}\pi r^3$ · 2: About 94 minutes · 3: $36\pi$ cm³

   • Half of $\frac{4}{3}\pi r^3$.
   • Step 1. $V=3.14\times 1\times 1.5=4.71$ m³ $=4710$ L.
     Step 2. $4710÷50\approx 94$ min.
   • $\frac{4}{3}\pi \times 27=36\pi$.

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