The sum of all interior angles of n-sided polygon is:

Question

Accepted Answer

The correct option is 1:
The standard formula for calculating the sum of all interior angles of an $n$-sided polygon is derived from the fact that any $n$-sided polygon can be divided into $n-2$ non-overlapping triangles by drawing diagonals from a single vertex. Since the sum of the interior angles of one triangle is $180^\circ$, the total sum is expressed as $(n-2) \times 180^\circ$. To see why option 1, $(2n-4) \times 90$, is correct, we must recognize that $180^\circ$ is equal to $2 \times 90^\circ$. Substituting this into the standard formula gives us $(n-2) \times (2 \times 90)$. By distributing the number 2 across the parenthesis, we get $2(n-2) \times 90$, which simplifies mathematically to $(2n - 4) \times 90$. This confirms that option 1 is a mathematically equivalent and correct representation of the interior angle sum formula.

The incorrect option is 2:
Option 2 presents the formula $(2n+4) \times 90$. If we factor out 2, this simplifies to $(n+2) \times 180^\circ$. This formula incorrectly assumes that an $n$-sided polygon can be divided into $n+2$ triangles, which is geometrically false. For a simple triangle ($n=3$), this formula would incorrectly yield $5 \times 180^\circ = 900^\circ$, which is far from the true sum of $180^\circ$. This expression is essentially adding $360^\circ$ to the correct sum and therefore provides an excessively large and inaccurate result for all polygons.

The incorrect option is 3:
Option 3 provides the expression $(n-2) \times 90$. This expression is exactly half of the correct total sum. For a 4-sided polygon (quadrilateral), the formula would give $(4-2) \times 90 = 180^\circ$, but we know the sum of angles in a quadrilateral must be $360^\circ$. The expression correctly identifies the $(n-2)$ triangles, but it incorrectly multiplies this number by $90^\circ$ instead of the full angle of $180^\circ$ present in each of those triangles.

The incorrect option is 4:
Option 4 suggests the calculation $(n+2) \times 90$. This expression is completely unrelated to the geometric principle used to determine the sum of interior angles. It is neither the correct sum nor any recognizable part of the correct calculation. For a triangle ($n=3$), this formula yields $(3+2) \times 90 = 5 \times 90 = 450^\circ$, which is mathematically incorrect, as the actual sum is $180^\circ$.