Scale Drawings: Areas and Reproduction

Lesson 2 of 2

In this lesson:

• Compute actual area from a scale drawing (area scales by k²!)
• Reproduce a drawing at a larger or smaller scale
• Solve multi-step real-world problems

What You Will Learn in Lesson 2

By the end of this lesson, you should be able to:

1. Compute the actual area of a figure from a scale drawing, recognizing that area scales by the square of the linear scale factor
2. Reproduce a given scale drawing at a new (larger or smaller) scale by computing new drawing lengths systematically
3. Solve multi-step real-world problems involving scale drawings

Quick Recap: Does Area Scale Like Length?

From Lesson 1: Lengths scale by k.

New question: If length and width both scale by k, does area also scale by k?

Think about a 1×1 square enlarged to a 2×2 square. Count the unit squares...

Area Grows Faster Than Length

Pattern: side doubles → area quadruples; side triples → area × 9

Algebra Confirms: Area Scales by k²

Linear factor: k → Area factor: k²

Worked Example: Park Rectangle Method 1

Scale: 1 cm:10 m. Drawing: 4 cm × 6 cm.

Method 1 — Convert sides, then area:

Worked Example: Park Rectangle Method 2

Same problem — verified using k²:

k (in matching units): 1 cm = 1000 cm → k = 1000

Use Method 1 as default; Method 2 verifies.

Wrong vs. Right: The Area Error

	Calculation	Result
Wrong	24 cm² × 10 = 240 m²	Off by ×10
✓ Right	40 m × 60 m = 2,400 m²	Correct

k applied once instead of k².

Check-In: Find the Actual Area

Scale: 1 cm:5 m. Drawing: 3 cm × 2 cm.

A) 30 m² B) 150 m² C) 300 m² D) 60 m²

Use Method 1 before choosing.

Worked Example: Area of a Triangular Plot

Scale: 1 cm:8 m. Triangle: base 5 cm, height 3 cm.

Convert both dimensions:

Compute area:

Your Turn: Solve Three Area Problems

Convert dimensions first. Show each step.

1. Scale 1 cm:6 m; rectangle 5 cm × 3 cm → area?
2. Scale 1 cm:4 m; triangle base 4 cm, height 2.5 cm → area?
3. Scale 2 cm:10 m; rectangle 6 cm × 4 cm → area?

Answers: Check Your Three Area Solutions

1. Scale 1 cm:6 m; 5×3 cm → actual 30×18 m → area = 540 m²
2. Scale 1 cm:4 m; base 4 cm, height 2.5 cm → actual 16×10 m → area = 80 m²
3. Scale 2 cm:10 m → 1 cm:5 m; 6×4 cm → actual 30×20 m → area = 600 m²

Transition: From Area to Reproduction

We can now compute actual area from any scale drawing.

New challenge: Take an existing drawing and redraw it at a different scale.

• Step 1 — Decode: use the original scale → find actual dimensions
• Step 2 — Encode: use the new scale → find new drawing dimensions

How the Two-Step Reproduction Works

• Step 1: Decode — multiply by old scale factor
• Step 2: Encode — divide by new scale factor

Worked Example: Reproduce the Garden Larger

Original: 6 cm × 4 cm at 1 cm:5 m. Reproduce at 1 cm:2 m.

Decode: m, m

Encode: cm, cm

New drawing (15×10 cm) is larger ✓

Worked Example: Same Garden Made Smaller

Same actual garden (30 m × 20 m). Reproduce at 1 cm:10 m.

Encode: cm, cm

New drawing (3×2 cm) is smaller ✓

1 cm:2 m → largest; 1 cm:5 m → medium; 1 cm:10 m → smallest

Scale Comparison Table for Same Garden

Misconception: Wrong Scale Used in Step 2

Error: Decodes correctly (×5 → 30 m), then encodes with OLD scale again (×5 → 150 cm).

Fix: Label each step explicitly:

• Step 1 (DECODE): ×old scale
• Step 2 (ENCODE): ÷new scale

Actual → drawing always divides.

Guided Practice: Reproduce the Hallway

Original: 3 cm × 0.5 cm at 1 cm:4 m. Reproduce at 1 cm:2 m.

Step 1 (given): m, m

Your turn — Step 2: Encode at 1 cm:2 m.

Find new dimensions. Larger or smaller than original?

Your Turn: Two Reproduction Problems

Label Step 1 (decode) and Step 2 (encode).

1. Drawing 4 cm × 2 cm at 1 cm:3 m → reproduce at 1 cm:6 m
2. Drawing 5 cm × 5 cm at 1 cm:8 m → reproduce at 1 cm:2 m

Predict larger or smaller before computing.

Answers: Check Your Reproduction Work

Problem 1: Actual 12 m × 6 m (×3). New drawing at 1 cm:6 m = 2 cm × 1 cm (smaller ✓)

Problem 2: Actual 40 m × 40 m (×8). New drawing at 1 cm:2 m = 20 cm × 20 cm (larger ✓)

Key Takeaways: Areas and Reproduction

✓ Length scales by k — area scales by k²

✓ Area: convert dimensions first, then compute

✓ Reproduction: decode (×old scale) → encode (÷new scale)

Never multiply drawing area by k

Convert every dimension (base AND height)

Step 2: use the new scale

Coming Up: Scale Drawings and Geometry

This standard connects forward to:

• 7.G.A.2: Constructing triangles and polygons at specified scales
• 7.G.B.6: Real-world area and volume problems
• 8.G.A: Similar figures and dilations
• High school: AA, SAS, SSS similarity theorems

Proportion is the backbone of all similarity work ahead.