← Unit 4 Measurement and Geometry Review

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MA5-ARE-C-01

Measurement and Geometry Review

⏱ 25 min📚 Year 9📈
Think First

Two similar cylinders have radii 3 cm and 6 cm. The smaller has volume 36$pi$ cm³. Find the larger volume without calculating from scratch.

💡 Revision: Ensure you understand area/volume formulas and scale factors.

Learning Intentions

Know

  • Surface area
  • Volume
  • Scale factor
  • Similarity
  • Congruence

Understand

  • How scale factors affect area and volume
  • When to use which formula

Can Do

  • Calculate SA and volume of prisms and cylinders
  • Apply scale factors to area and volume
  • Prove congruence and similarity
Scale factorSAVolumeSimilarCongruent
Learn Phase
1

Area and Volume Formulas

Quick reference

Key formulas:

  • Rectangular prism: $V = lwh$, $SA = 2(lw + lh + wh)$
  • Cylinder: $V = pi r^2 h$, $SA = 2pi r^2 + 2pi r h$
  • Any prism: $V = A_{ ext{cross}} imes h$

For composite solids: decompose, calculate each part, add/subtract.

2

Scale Factors

Square for area, cube for volume

If lengths scale by factor $k$:

Scale Relationships

$ ext{Area} ightarrow k^2$

$ ext{Volume} ightarrow k^3$

Example: Scale factor 2. If original area = 10, new area = 40.

If original volume = 8, new volume = 64.

3

Congruence and Similarity

Proving relationships

Congruence tests: SSS, SAS, ASA, RHS

Similarity tests: SSS (sides in proportion), SAS (two sides proportional, included angle equal), AA (two angles equal)

Remember: AAA proves similarity, NOT congruence.

✓

Check Understanding

Try it yourself

A model building is 1:50 scale. If the model volume is 0.4 m³, what is the actual volume?

Worked Example

Measurement Review

1

A cylinder has radius 4 cm, height 10 cm. Find volume and surface area.

$V = pi(4)^2(10) = 160pi approx 502.7$ cm³

$SA = 2pi(16) + 2pi(4)(10) = 32pi + 80pi = 112pi approx 351.9$ cm²

2

Two similar cones have heights 5 and 15. The smaller has volume $20pi$. Find the larger volume.

$k = 15/5 = 3$

$V_{ ext{large}} = 3^3 imes 20pi = 27 imes 20pi = 540pi$

3

Prove that two equilateral triangles with sides 4 and 8 are similar but not congruent.

All angles = 60°, so similar by AA.

Sides are in ratio 1:2, not equal, so not congruent.

Common Misconceptions

✕

Apply linear scale factor to area. No — area scales as $k^2$. If lengths double, area quadruples.

✕

Apply area scale factor to volume. No — volume scales as $k^3$. If lengths double, volume increases 8-fold.

✕

Use AAA for congruence. AAA proves similarity only. Triangles with the same angles can be different sizes.

Your Turn

Practice — Measurement Review

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

1Find the volume of a composite solid: rectangular prism 6×5×4 with half-cylinder (r=2, h=6) on top.
2Two similar pyramids have base areas 16 and 64. Find the ratio of their volumes.
3A rectangular tank (2m×1.5m×1m) is half full. How many litres?
Real-World Anchor

Architecture and Engineering

The Sydney Opera House required extensive scale modelling before construction. Architects calculated how surface area and volume would scale from model to full size to estimate materials, costs, and structural requirements.

📓 Copy Into Your Books

▼

Formulas

  • Cylinder: $V=pi r^2 h$
  • Prism: $V=A_{ ext{cross}} imes h$
  • SA = sum of all faces

Scale

  • Length: $k$
  • Area: $k^2$
  • Volume: $k^3$

Tests

  • Congruence: SSS, SAS, ASA, RHS
  • Similarity: SSS, SAS, AA
  • AAA ≠ congruence
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
Scale factor 2. Area becomes:
Scale factor 3. Volume becomes:
Cylinder radius 3, height 4. Volume:
Model 1:100. Real volume is model volume ×
Two similar solids, sides 2:5. Volume ratio:
AAA proves:
Rectangular prism 4×3×2. SA:
Composite solid: add volumes when:
1A water tank is a cylinder (r=1.5 m, h=3 m). Find its capacity in litres.
2Two similar spheres have radii 2 and 6. The smaller has SA = 16$pi$. Find the larger SA and volume.
3Explain why two rectangles with sides 2×3 and 4×6 are similar.

Comprehensive Answers

1Cylinder r=1.5, h=3. Litres?
$V = pi(2.25)(3) = 6.75pi$ m³ ≈ 21.2 m³ = 21,200 L.
2Spheres radii 2 and 6, SA=16$pi$.
$k=3$, SA = $9 imes 16pi = 144pi$. V = $27 imes (32pi/3) = 288pi$.
3Why 2×3 and 4×6 similar.
Sides in ratio 1:2, all angles 90°. Similar by SSS or SAS.
MC 1Scale 2, area.
$2^2 = 4$. Answer: B
MC 2Scale 3, volume.
$3^3 = 27$. Answer: D
MC 3Cylinder volume.
$pi(9)(4) = 36pi$. Answer: C
MC 4Model 1:100, volume.
$100^3 = 1,000,000$. Answer: C
MC 5Sides 2:5, volume ratio.
$2^3:5^3 = 8:125$. Answer: C
MC 6AAA proves.
Similarity. Answer: B
MC 7Prism 4×3×2 SA.
$2(12+8+6) = 52$. Answer: C
MC 8Add volumes when.
Joining parts. Answer: A
SA 1Tank capacity.
21,200 L.
SA 2Spheres SA and V.
SA = 144$pi$, V = 288$pi$.
SA 3Similar rectangles.
Sides in proportion, angles equal.
Game Phase
🎲
Game Unlocked!
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📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.