The interior angles of a polygon with n sides, are in an A.P. with common difference 6°. If the larg

Q: Wait, why do we reject the negative root $n = -6$?

That is a fantastic question! In mathematics, quadratic equations often yield two solutions, but in physics and geometry, we must apply physical constraints. Since $n$ represents the number of sides of a polygon, it must be a positive integer ($n > 2$). A polygon cannot have a negative number of sides, so we discard $n = -6$ as an extraneous solution.

Q: Why can't we just use the sum of angles formula alone?

The sum of angles formula, $(n-2) \times 180^\circ$, is indeed our anchor, but it contains two unknowns: the sum itself and the number of sides $n$. By using the A.P. property, we create a second, independent equation involving $n$. This system of two equations allows us to solve for the single unknown $n$.

Q: Is it possible for an interior angle to be $219^\circ$? I thought interior angles were always less than $180^\circ$.

You are thinking of convex polygons! A convex polygon has all interior angles less than $180^\circ$. However, the problem does not specify that the polygon is convex. A polygon with an interior angle greater than $180^\circ$ is called a concave polygon. The math holds true regardless of convexity, so $219^\circ$ is perfectly valid here.

Question

The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{\circ}.$ If the largest interior angle of the polygon is $219^{\circ}$, then n is equal to

Sigma Percentile · Accepted Answer

### The Geometric Anchor: Understanding the Polygon's Soul

Imagine you are standing in the center of a mysterious, $n$-sided polygon. You don't know how many sides it has, but you know one fundamental truth: the sum of its interior angles is governed by a rigid law of geometry.

This law states that the sum of all interior angles $S_n$ is given by the elegant expression:

$$S_n = (n-2) 	imes 180^{\circ}$$

This is our geometric anchor, the bedrock upon which we build our solution. It remains constant for any $n$-sided shape, whether regular or irregular, convex or concave.

### The A.P. Connection: Unlocking the Sequence

The problem introduces a layer of complexity by stating the interior angles follow an Arithmetic Progression (A.P.). We are given the common difference $d = 6^{\circ}$ and the largest angle (the $n$-th term) $l = 219^{\circ}$.

To find the sum of these angles using the A.P. formula, we first express the first term $a$ in terms of $n$. We know the $n$-th term is defined as:

$$l = a + (n-1)d$$

Substituting our known values:

$$219 = a + (n-1)6$$

Rearranging to solve for $a$:

$$a = 219 - 6(n-1) = 225 - 6n$$

### The Bridge: Equating the Two Worlds

We now have two different ways to describe the sum of the angles. We equate the geometric sum to the A.P. sum formula, $S_n = \frac{n}{2}(a + l)$:

$$(n-2) 	imes 180 = \frac{n}{2}(a + l)$$

Substituting our expression for $a$ into this bridge equation:

$$(n-2) 	imes 180 = \frac{n}{2}(225 - 6n + 219)$$

Simplifying the right side:

$$(n-2) 	imes 180 = \frac{n}{2}(444 - 6n)$$

$$(n-2) 	imes 180 = n(222 - 3n)$$

### The Quadratic Resolution: Finding $n$

Expanding both sides of the equation, we obtain:

$$180n - 360 = 222n - 3n^2$$

Rearranging the terms into a standard quadratic form:

$$3n^2 - 42n - 360 = 0$$

Dividing the entire equation by $3$ yields:

$$n^2 - 14n - 120 = 0$$

Factoring the quadratic equation:

$$(n - 20)(n + 6) = 0$$

This results in two potential values: $n = 20$ or $n = -6$. Since a polygon cannot have a negative number of sides, we reject $n = -6$.

The number of sides of the polygon is **$n = 20$**.

Animated Solution for Mathematics - Sequence and Series: The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{\circ} .$ If the largest interior angle of the polygon is $21 9^{\circ}$ , then n is equal to

Enter Numerical Value:

Visualized Solution

Geometric Sum of Polygon Angles

Angles in Arithmetic Progression

Finding the First Term $a$

Simplifying the First Term

Arithmetic Sum Formula

Substituting into A.P. Sum

Simplifying the A.P. Sum

Equating the Two Sums

Expanding the Equation

Rearranging to Standard Form

Simplifying the Quadratic Equation

Factorizing the Quadratic

Final Answer and Conclusion

The Sigma Insight: Arithmetic Progression (A.P.)

The Geometric Anchor

The A.P

The Bridge

The Quadratic Resolution

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