IPMAT Indore Polygons — PYPs

The Setup: A pure geometry fundamental. The sum of the interior angles of any convex $n$-sided polygon is the GOAT formula $(n-2)\times 180^\circ$. Set up the inequality, solve, and take the largest integer that survives.
Step 1: Set up the inequality. The sum is strictly less than $2019^\circ$:

$$(n-2)\times 180 < 2019$$

Step 2: Isolate the variable. Divide both sides by 180 (positive, so the inequality direction is safe):

$$n-2<\frac{2019}{180}$$

Step 3: Crunch the division. Since $180\times 11=1980$ and $180\times 12=2160$, the quotient sits between 11 and 12:

$$\frac{2019}{180}=11.216\ldots \implies n<13.216\ldots$$

Step 4: Lock in the maximum integer. A polygon has a whole number of sides, so the largest admissible $n$ is 13.
Step 5: Verify both sides of the boundary. A bound is only trustworthy if the next value up actually fails:
* $n=13$: sum $=(13-2)\times 180=1980^\circ$, and $1980<2019$. Valid.
* $n=14$: sum $=(14-2)\times 180=2160^\circ$, and $2160>2019$. Fails.

So 13 works and 14 does not, which pins the maximum exactly.

Final Answer: 13

The Setup: We are dropping into a geometry raid involving a regular octagon inscribed inside a circle. Think of it like slicing up a pizza into 8 perfectly equal pieces. To find the total area, we can calculate the area of just one of those triangular pizza slices (sectors) branching out from the center, and then multiply it by 8.

Step 1: Analyze the central angles of the octagon.
A full circle is $360^\circ$ (or $2\pi$ radians). Since a regular octagon has $8$ identical sides, it also has $8$ identical triangles radiating out from the center point to each vertex.
Let's find the central angle $\theta$ for just one of these triangles:
$$\theta = \frac{360^\circ}{8} = 45^\circ$$

Step 2: Pull the stats for a single triangle.
Each of these $8$ triangles shares two sides that extend from the center to the circle's boundary. Since both of those lines are radii of the circle, their lengths are both equal to $r = 1$.
So, we have an isosceles triangle with two side lengths of $1$ and an included angle of $\theta = 45^\circ$.

Step 3: Calculate the area of one triangle.
The trigonometry cheat code for finding the area of a triangle when you know two sides ($a$ and $b$) and the included angle ($\theta$) is:
$$\text{Area}_{\text{triangle}} = \frac{1}{2} a b \sin(\theta)$$
Plug in our stats ($a = 1$, $b = 1$, and $\theta = 45^\circ$):
$$\text{Area}_{\text{triangle}} = \frac{1}{2} (1)(1) \sin(45^\circ)$$
Since $\sin(45^\circ) = \frac{\sqrt{2}}{2}$:
$$\text{Area}_{\text{triangle}} = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4}$$

Step 4: Scale it up to the full octagon.
An octagon consists of $8$ of these exact identical triangles. To get the total area, just multiply the area of one triangle by $8$:
$$\text{Total Area} = 8 \cdot \left(\frac{\sqrt{2}}{4}\right)$$
Simplify the numbers:
$$\text{Total Area} = 2\sqrt{2}$$
The math is clean, fast, and flawless.

Final Answer: $2\sqrt{2}$