LEARN · EXPLAIN · REVISE

Under a half-depth 45° oblique drawing, why is the horizontal area scale √2/4?

Read the idea, work independently, then explain what changed.

高一必修第二册(A版).pdf · 8.2 · PDF 114 / printed page 107

TOPIC 01

Oblique drawings of solids

Build understanding of oblique drawings of solids through definitions, contrasting cases and justified applications.

Use definitions, valid spatial reasoning and metric checks for oblique drawings of solids.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Compare a rectangular area with the corresponding drawn parallelogram.

Hint 2

Area depends on both adjacent lengths and their included angle.

Worked solution

Compare a rectangular area with the corresponding drawn parallelogram.
Calculate or simplify this relation.
$\frac{ab(1/2)\sin45^{\circ}}{ab}=\sqrt2/4$
The ratio applies to the stated horizontal drawing convention.

The depth scale contributes 1/2 and the oblique angle contributes sin45°=√2/2.

Checks and common pitfalls: The ratio applies to the stated horizontal drawing convention.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Use definitions, valid spatial reasoning and metric checks for oblique drawings of solids.
Drawing convention: The common 45° oblique convention halves receding lengths while retaining other stated scales.
Representation limits: A sketch preserves selected incidence and parallel relations, but not all angles or distances.
Close with: conditions → representation → reasoning → check.

Expected reasoning: The common 45° oblique convention halves receding lengths while retaining other stated scales.
Expected reasoning: A sketch preserves selected incidence and parallel relations, but not all angles or distances.
Expected correction: A measured sketch length need not equal a true spatial length.