GANITA MANJARI · GRADE 9 · PART I · CHAPTER 6
PUZZLE — Perimeter Reasoning
Think first. Use AI purposefully. Verify always.
DISCOVER
Start with a question.
Can several curved pieces make a path exactly as long as one large semicircle?
Say what you expect to stay the same and what you expect to change. Then test your prediction in the visual below.
Understand
Semicircular arc length is proportional to diameter. Interior dotted arcs are not part of the outer boundary of a union of circles.
Key Ideas
More pieces do not necessarily mean a longer path. Do not include internal arcs in an outer perimeter.
Visualise
Compare one semicircular arc with two arcs whose diameters partition the same segment. Ignore the straight baseline.
The path lengths are equal, even though the shapes differ.
Method
Mark the boundary being measured. Express each arc as a fraction of its circumference, then use relationships between diameters or central angles.
Worked Example
Given
Two circles of radius r pass through one another’s centres (Example 1).
To Find
Find the perimeter of their union.
Formula / Property / Theorem / Reasoning
The triangles from the two centres to either intersection are equilateral.
Solution
- Each internal arc subtends 120°. Each outer arc therefore subtends 240°.
- Each outer arc has length () × 2πr = .
- Add the two outer arcs.
Result
units.
AIMS Method
- A — Attempt independently. Make a prediction or start a sketch.
- I — Investigate the idea. Identify the boundary, region and given conditions.
- M — Make connections. Relate the figure to a known property or rearrangement.
- S — Scrutinise your reasoning. Check conditions, units and whether the result is exact.
Think first. Use AI purposefully. Verify always.
Focused Practice
Two semicircles have diameters 4 cm and 6 cm. Compare their combined arc length with a semicircle of diameter 10 cm.
Check & Feedback
Why must the words “outer boundary” be checked before adding arcs?
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.5 · Examples 1–2, Figures 6.12–6.13 · printed pages 127–128.