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Polygons & Ratios

Scale factors and angle sums. The rules of shape manipulation.

Best Geometry Hack

Geometry

\[ \text{Area Ratio} = (\text{Side Ratio})^2 \]

If Side A is 3x bigger than Side B, Area A is \(3^2 = 9x\) bigger. Don't fall for the linear trap.

📝 Example: Square scaling

Triangle A and B are similar. Side ratio is 2:5. Area of A is 20. Area of B?

\(\text{Area Ratio} = (2/5)^2 = 4/25\)

\(\frac{20}{B} = \frac{4}{25} \rightarrow 4B = 500 \rightarrow B = 125\)

Result: 125.

Geometry

\[ \text{Sum} = (n-2) \cdot 180^\circ \]

For any polygon with \(n\) sides.

📝 Example: Pentagon Sum

Sum of angles in a 5-sided shape (pentagon)?

\((5-2) \cdot 180 = 3 \cdot 180 = 540^\circ\)

Geometry

Always \(360^\circ\)

No matter how many sides a convex polygon has, the exterior angles sum to 360.

📝 Example: The 360 Rule

Whether it's a triangle or a 100-sided polygon, walking around the exterior always completes one full revolution (\(360^\circ\)).