THE GEOMETRY OF POLYGONS

$$(6-2)\times180=720°$$

$$720\div6=120°$$

$$(9-2)\times180=1260°$$

$$1260\div9=140°$$

Hexagon: 4 triangles. Heptagon: 5 triangles — the number of triangles is always $n-2$.

From one vertex, you can draw a diagonal to every OTHER vertex except the two adjacent to it (since those would just be the polygon's existing sides, not diagonals) and itself — this creates $n-3$ diagonals, which divide the polygon into $n-2$ triangles.

Since each triangle contributes $180°$ to the total angle sum, and there are always $(n-2)$ triangles, the total angle sum must be $(n-2)\times180°$.

The student's formula has no real mathematical basis — the correct formula is $\frac{(n-2)\times180}{n}$ (angle sum divided by number of sides). Correct hexagon interior angle: $\frac{(6-2)\times180}{6}=\frac{720}{6}=120°$, not 174°.

Student's formula for $n=3$: $180-3=177°$ — but a real EQUILATERAL triangle should have interior angles of exactly $60°$ each, so $177°$ is clearly wrong. Testing a formula against a simple, well-known case (like an equilateral triangle) is a quick, reliable way to catch an incorrect formula before using it further.

$$\frac{(12-2)\times180}{12}=\frac{1800}{12}=150°$$

$$\frac{(n-2)\times180}{n}=160 \Rightarrow 180n-360=160n \Rightarrow 20n=360 \Rightarrow n=18 \text{ sides}$$

$$(14-2)\times180=2160°$$

$$2160\div14\approx154.3°$$

Sum: $(11-2)\times180=1620°$. Each: $1620\div11\approx147.3°$.

Interior: $\frac{(6-2)\times180}{6}=120°$. Exterior: $180-120=60°$.

Nonagon: interior $=140°$, exterior $=40°$. Dodecagon: interior $=150°$, exterior $=30°$. Pattern: exterior angle $=\dfrac{360°}{n}$ directly — verify: hexagon $360\div6=60°$?, nonagon $360\div9=40°$?, dodecagon $360\div12=30°$?.

Sum of exterior angles $= n\times180° - \text{(sum of interior angles)} = 180n - (n-2)\times180 = 180n - 180n + 360 = 360°$.

If you walk all the way around ANY closed polygon's perimeter, returning to your starting point facing the SAME direction you began, the TOTAL amount you've turned (the sum of all the exterior 'turning' angles at each corner) must equal exactly one full rotation, $360°$ — this is true regardless of how many sides the polygon has, which is why the exterior angle sum is always 360° universally.

The formula requires subtracting 2 from $n$ BEFORE multiplying by 180 (since the polygon splits into $(n-2)$ triangles, not $n$ triangles) — the student forgot this subtraction. Correct: $(8-2)\times180=1080°$.

$(4-2)\times180=360°$ ? — matches the well-known fact that a quadrilateral's angles sum to 360°, confirming the formula $(n-2)\times180$ is correct.

The angle SUM formula depends only on $n$ (the number of sides), because ANY simple polygon with $n$ sides — regular or irregular — can always be divided into exactly $(n-2)$ triangles from one vertex, each contributing 180° to the total. What DOES differ between regular and irregular polygons is how that TOTAL is DISTRIBUTED among the individual angles (equal in a regular polygon, unequal in an irregular one) — but the total itself is always the same for a given $n$.

$$\frac{(n-2)\times180}{n}=156 \Rightarrow 180n-360=156n \Rightarrow 24n=360 \Rightarrow n=15$$

$\frac{(n-2)\times180}{n}=168 \Rightarrow 180n-360=168n \Rightarrow 12n=360 \Rightarrow n=30$ sides. As the interior angle approaches 180° (requiring more and more sides), a regular polygon increasingly resembles a smooth CIRCLE, rather than a sharp, star-like or angular shape — more sides with larger interior angles create a rounder, smoother overall appearance.

$$\frac{(9-2)\times180}{9}=140°$$

A regular nonagon, having 9 relatively 'rounded' sides approximating a circle-like shape, cannot perfectly fill a square's sharp 90° CORNERS — the nonagon's rounded outline means there will always be small triangular/curved gaps of wasted material near each of the square's four corners. Since a nonagon (with many sides) closely approximates a circle, and a circle inscribed in a square wastes about $1-\frac{\pi}{4}\approx21.5\%$ of the square's area, the nonagon's wasted area is likely to be a MODERATE (not tiny, not huge) percentage — roughly in a similar ballpark to this circle estimate, though slightly less wasteful than a true circle since the nonagon's corners extend slightly further into the square's corner regions than a perfect circle would.