← Unit 4 Composite Solids

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Composite Solids

⏱ 25 min📚 Year 9📈
Think First

A building consists of a rectangular prism topped by a triangular prism. How would you find its total volume?

💡 Revision: Ensure you can find volume and surface area of prisms and cylinders.

Learning Intentions

Know

  • Decomposition
  • Adding volumes
  • Subtracting volumes
  • Hidden faces

Understand

  • Why some faces disappear when solids are joined
  • How to avoid double-counting areas

Can Do

  • Decompose any composite solid
  • Calculate total volume
  • Calculate total surface area accounting for hidden faces
DecomposeHidden faceOverlapNet
Learn Phase
1

Decomposing Solids

Break it down

To find the volume of a composite solid:

  1. Identify the simple shapes that make it up
  2. Calculate each volume
  3. Add them together

Example: A house = rectangular prism + triangular prism.

$V_{ ext{total}} = V_{ ext{rect}} + V_{ ext{tri}}$

2

Surface Area of Composites

Watch for hidden faces

When two solids are joined, some faces are hidden (internal) and should not be counted in surface area.

Strategy:

  1. Calculate SA of each part separately
  2. Identify the overlapping (hidden) faces
  3. Subtract twice the overlap area

Example: Two cubes joined face-to-face: subtract $2 imes ext{face area}$.

3

Subtractive Composites

Solids with holes

Some composites are formed by removing material:

  • A rectangular block with a cylindrical hole
  • A sphere with a conical indentation

Volume = Volume of original − Volume removed

Surface area = SA of original + SA of hole (inner surface) − 2 × area of hole opening

✓

Check Understanding

Try it yourself

A solid consists of a cylinder (r=3, h=4) on top of a cube (side 6). Find the total volume.

Worked Example

Composite Solids

1

A box 10×8×6 has a cylindrical hole (r=2, h=6) drilled through it. Find remaining volume.

$V_{ ext{box}} = 10 imes 8 imes 6 = 480$

$V_{ ext{hole}} = pi imes 4 imes 6 = 24pi approx 75.4$

$V_{ ext{remaining}} = 480 - 75.4 = 404.6$ cm³

2

Two cubes (side 4) are joined face-to-face. Find total SA.

SA of two separate cubes: $2 imes 6(4)^2 = 192$

Hidden faces: $2 imes 16 = 32$

Total SA = $192 - 32 = 160$ cm²

3

A prism (rectangular 5×4×3) has a half-cylinder (r=2, h=5) removed from one end. Find volume.

$V_{ ext{prism}} = 5 imes 4 imes 3 = 60$

$V_{ ext{half-cyl}} = rac{1}{2} imes pi imes 4 imes 5 = 10pi approx 31.4$

$V = 60 - 31.4 = 28.6$ cm³

Common Misconceptions

✕

Count hidden faces in surface area. When two solids join, the touching faces are not part of the external surface.

✕

Forget to subtract the hole opening from both sides. A hole through a solid removes material from two faces.

✕

Add instead of subtract for removed material. If a hole is drilled, subtract its volume from the total.

Your Turn

Practice — Composite Practice

Work through each question in your book or digitally. Answers are in the Questions phase.

1A solid is a cylinder (r=4, h=5) with a cone (r=4, h=3) on top. Find total volume.
2Two rectangular prisms (5×4×3 and 5×4×2) are stacked. Find total SA.
3A sphere (r=5) has a cylindrical hole (r=2, h=10) through its centre. Find remaining volume.
Real-World Anchor

Architecture and Sculpture

The Sydney Opera House and modern sculptures are composite solids. Architects decompose complex buildings into simple geometric forms to calculate material quantities, structural loads, and heating/cooling requirements.

📓 Copy Into Your Books

▼

Volume

  • Decompose into parts
  • Add volumes
  • Subtract holes

Surface Area

  • Calculate each part
  • Subtract hidden faces ×2
  • Add inner surfaces for holes

Check

  • Does answer make sense?
  • Units correct?
  • Did you account for all faces?
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
Two cubes (side 3) joined face-to-face. Total SA:
A hole is drilled through a solid. For volume, you:
When joining two solids, hidden faces:
A composite solid SA calculation must account for:
Cube side doubled. Surface area becomes:
A rectangular block (6×4×3) with cylindrical hole (r=1, h=6). Remaining volume ≈
For a hole through a solid, the inner surface:
The best first step for composite solids is:
1A solid is a rectangular prism (8×6×4) with a half-cylinder (r=3, h=8) on top. Find total volume.
2Two cubes (side 5) are joined along one face. Find the total surface area.
3Explain why you subtract twice the overlap area when finding SA of joined solids.

Comprehensive Answers

1Prism 8×6×4 + half-cylinder r=3, h=8.
$V = 192 + rac{1}{2}pi(9)(8) = 192 + 36pi approx 305.1$ cm³.
2Two cubes side 5 joined.
$2 imes 150 - 2(25) = 300 - 50 = 250$ cm².
3Why subtract 2× overlap.
Each solid loses one face. Total loss = 2 × face area.
MC 1Two cubes side 3 joined.
$2(54) - 2(9) = 108 - 18 = 90$. Answer: B
MC 2Hole volume.
Subtract. Answer: B
MC 3Hidden faces.
Not counted. Answer: C
MC 4SA accounts for.
Only outer faces. Answer: A
MC 5Cube doubled.
$2^2 = 4$. Answer: B
MC 6Block with hole.
$72 - pi(1)^2(6) = 72 - 6pi$. Answer: A
MC 7Inner surface.
Added to SA. Answer: B
MC 8Best first step.
Decompose. Answer: B
SA 1Prism + half-cylinder.
$192 + 36pi approx 305.1$ cm³.
SA 2Two cubes side 5.
$250$ cm².
SA 3Subtract 2× overlap.
Each solid loses one face.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.