EQN-06-04

Comparison with Geometry & Figures

Compare areas, perimeters, and angles using relationships — and never trust a figure that may not be to scale.

~6 min read•Difficulty 3/5

Never measure the drawing

In a comparison question, a figure is not drawn to scale unless stated. Reason from the given numbers and geometric rules, never from how long or wide something looks.

Compare by relationship, not by computing

Often you can decide without full values.

Quantity A : area of a square with side 4

Quantity B : area of a circle with radius 2

A = 16, B = 4 π \approx 12.57 \to A is greater

Leaving B as $4 π$ and knowing $π \approx 3.14$ settles it without a calculator.

Same perimeter $\neq =$ same area

Among shapes with equal perimeter, the more "compact" one (a circle, then a square) has the larger area. A long thin rectangle has far less area than a square of the same perimeter.

When it's "cannot be determined"

If the figure leaves a length or angle unfixed, multiple configurations are possible — and that means choice D. Test whether the given facts pin the shape down completely before deciding.

Worked examples

Example 1 (difficulty 2/5)

Quantity A: the perimeter of a square of area 36. Quantity B: 25. Which is greater?

◎ Predict the answer first:

Example 2 (difficulty 3/5)

A triangle has two sides of length 5 and 9. Quantity A: the length of the third side. Quantity B: 8. Which is greater?

◎ Predict the answer first:

Tips

Treat every figure as "not to scale" — reason from numbers and rules only.
Leave $π$ in your work $(π \approx 3.14)$ so circle comparisons need no calculator.
If a length or angle is free to vary, the answer is usually "cannot be determined".

Common mistakes

Judging which segment or angle is larger by eye

Why: The drawing looks authoritative.

Fix: Comparison figures are not to scale; use the given measures and theorems.

Assuming equal perimeter means equal area

Why: Both feel like "size".

Fix: Compactness matters: a square beats a thin rectangle of the same perimeter.

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