If a regular polygon has 35 diagonals, then the sum of its interior angles is:

Question

Examoogle · Accepted Answer

The correct answer is Option B: 1440°. Test standard polygon values: for decagon ($n=10$), diagonals = $10 	imes 7 / 2 = 35$. Then immediately calculate sum = $(10 - 2) 	imes 180^{\circ} = 8 	imes 180^{\circ} = 1440^{\circ}$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Test standard polygon values: for decagon ($n=10$), diagonals = $10 	imes 7 / 2 = 35$. Then immediately calculate sum = $(10 - 2) 	imes 180^{\circ} = 8 	imes 180^{\circ} = 1440^{\circ}$.

Formula Used: $$d = \frac{n(n - 3)}{2}, \quad S = (n - 2) 	imes 180^{\circ}$$

Given: Number of diagonals of a regular polygon = 35

Steps: Step 1: Understand the formula for diagonals: The total number of diagonals $d$ in a polygon with $n$ sides is given by the formula: $$d = \frac{n(n-3)}{2}$$. Step 2: Solve for the number of sides $n$: Substitute $d = 35$ into the diagonal formula: $$\frac{n(n-3)}{2} = 35 \implies n(n-3) = 70$$ By inspection or solving the quadratic equation $n^2 - 3n - 70 = 0$, we get $(n-10)(n+7) = 0$, which yields $n = 10$.. Step 3: Calculate the sum of interior angles: The sum of interior angles of an $n$-sided polygon is given by the formula: $$S = (n - 2) 	imes 180^{\circ}$$. Step 4: Evaluate the final sum: Substitute $n = 10$ into the sum formula: $$S = (10 - 2) 	imes 180^{\circ} = 8 	imes 180^{\circ} = 1440^{\circ}$$

Answer: B is correct because the polygon has 10 sides ($n=10$) which gives an interior angle sum of $1440^{\circ}$.

Exam Tip: Knowing standard polygons like decagon ($n=10$), pentagon ($n=5$), and hexagon ($n=6$) saves crucial time in railway and civil service exams.

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