Polygons and Proportionality

TL;DR

Polygons are closed shapes made of straight lines, and you can classify them by their sides and angles. Proportionality describes how two shapes are related when one is just a scaled version of the other. Understanding these helps you compare sizes and solve for unknown measurements in similar figures.

1. The Mental Model

Think of polygons as building blocks in geometry, like LEGO pieces. Proportionality is like zooming in or out on a picture – the picture stays the same, just bigger or smaller.

2. The Core Material

What's a Polygon?

A polygon is a flat, 2D shape with straight sides that's completely closed. No curves, no open ends! The points where the sides meet are called vertices.

You classify polygons by the number of sides they have:
* 3 sides: Triangle
* 4 sides: Quadrilateral
* 5 sides: Pentagon
* 6 sides: Hexagon
* 8 sides: Octagon
* ...and so on!

A regular polygon has all sides equal in length AND all angles equal in measure. Think of a perfect square or an equilateral triangle.

Interior and Exterior Angles

• Interior angles are the angles inside the polygon. For any polygon with 'n' sides, the sum of its interior angles is (n - 2) * 180°.
• Exterior angles are formed by extending one side of the polygon. Each exterior angle and its adjacent interior angle always add up to 180°. The sum of all exterior angles of any convex polygon is always 360°.

Here's how to think about classifying polygons:

graph TD
    A["Shape"] --> B["Is it 2D and closed?"]
    B -- "No" --> C["Not a Polygon"]
    B -- "Yes" --> D["Does it have only straight sides?"]
    D -- "No" --> C
    D -- "Yes" --> E["Polygon"]
    E --> F["How many sides?"]
    F -- "3" --> G1["Triangle"]
    F -- "4" --> G2["Quadrilateral"]
    F -- "5" --> G3["Pentagon"]
    F -- "6" --> G4["Hexagon"]
    F -- "n" --> Gn["n-gon"]
    E --> H["Are all sides equal AND all angles equal?"]
    H -- "Yes" --> I["Regular Polygon"]
    H -- "No" --> J["Irregular Polygon"]

Proportionality and Similar Polygons

Proportionality comes into play when you have similar polygons. Two polygons are similar if:
1. All their corresponding angles are equal.
2. All their corresponding sides are proportional. This means the ratio of any pair of corresponding sides is constant. This constant ratio is called the scale factor.

Imagine you have a small triangle and a big triangle. If they're similar, the small one is just a perfectly scaled-down version of the big one. All angles match, and if one side in the big triangle is twice as long as the corresponding side in the small one, then all sides in the big triangle will be twice as long as their corresponding sides in the small one.

Let's say you have two similar triangles, $\triangle ABC$ and $\triangle DEF$.
* Angle A = Angle D
* Angle B = Angle E
* Angle C = Angle F
* $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = \text{scale factor}$

This scale factor is super useful because if you know it, and you know one side length in one polygon, you can find the corresponding side length in the similar polygon.

3. Worked Example

You have two quadrilaterals. Quadrilateral ABCD is similar to Quadrilateral EFGH.
Side AB = 6 cm, BC = 9 cm, CD = 12 cm.
Side EF = 2 cm, FG = 3 cm.
We need to find the length of side GH and the scale factor from ABCD to EFGH.

1. Identify corresponding sides:
   AB corresponds to EF.
   BC corresponds to FG.
   CD corresponds to GH.

2. Calculate the scale factor (from ABCD to EFGH):
   We can use the known corresponding sides: AB and EF.
   Scale factor = $\frac{\text{ side length in EFGH }}{\text{ corresponding side length in ABCD }}$
   Scale factor = $\frac{EF}{AB} = \frac{2 \text{ cm}}{6 \text{ cm}} = \frac{1}{3}$

   Let's check with another pair: BC and FG.
   Scale factor = $\frac{FG}{BC} = \frac{3 \text{ cm}}{9 \text{ cm}} = \frac{1}{3}$
   This confirms our scale factor is $\frac{1}{3}$.

3. Find the unknown side GH:
   CD corresponds to GH. We know CD = 12 cm and the scale factor is $\frac{1}{3}$.
   $\frac{GH}{CD} = \text{scale factor}$
   $\frac{GH}{12 \text{ cm}} = \frac{1}{3}$
   To find GH, multiply both sides by 12 cm:
   $GH = 12 \text{ cm} \times \frac{1}{3}$
   $GH = 4 \text{ cm}$

So, the length of side GH is 4 cm and the scale factor from ABCD to EFGH is $\frac{1}{3}$.

4. Key Takeaways

• Polygons are closed, 2D shapes made only of straight line segments.
• The sum of interior angles of an n-sided polygon is found using (n-2) * 180°.
• The sum of exterior angles of any convex polygon is always 360°.
• Similar polygons have corresponding angles that are equal.
• Similar polygons have corresponding sides that are proportional, linked by a constant scale factor.
• You use the scale factor to find unknown side lengths in similar figures.
• Make sure you always compare corresponding sides when setting up proportions.

5. Now Try It

You have a rectangular photo that is 10 inches wide and 8 inches tall. You want to print a smaller, similar version for your wallet. If the wallet photo needs to be 2.5 inches wide, what will its height be? What is the scale factor from the original photo to the wallet photo?