All Exams Test series for 1 year @ ₹349 only
Question

If the sum of all interior angles of a regular polygon is twice the sum of all its exterior angles then the polygon is

This question was previously asked in
CDS I 2020 Elementary Mathematics Previous Year Paper (02-Feb-2020)
The correct answer is

Hexagon

Understanding Regular Polygon Angle Properties

This question involves understanding the properties of regular polygons, specifically the relationship between their interior and exterior angles. A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

Formulas for Polygon Angles

To solve this problem, we need two key formulas:

  • Sum of Interior Angles: For any convex polygon with n sides, the sum of the interior angles is given by the formula: \[ S_i = (n-2) \times 180^\circ \] Here, '\(n\)' represents the number of sides of the polygon.
  • Sum of Exterior Angles: The sum of the exterior angles (one at each vertex) of any convex polygon, including regular polygons, is always constant and equal to: \[ S_e = 360^\circ \]

Setting Up the Equation for the Polygon

The question states a specific condition: "the sum of all interior angles of a regular polygon is twice the sum of all its exterior angles". We can translate this statement into a mathematical equation using the formulas above.

Given condition: \(S_i = 2 \times S_e\)

Substituting the formulas:

\[ (n-2) \times 180^\circ = 2 \times 360^\circ \]

Calculating the Number of Sides

Now, we need to solve this equation to find the value of '\(n\)', which tells us the number of sides the polygon has.

  1. Simplify the right side: Calculate the value of \(2 \times 360^\circ\). \[ 2 \times 360^\circ = 720^\circ \] So the equation becomes: \[ (n-2) \times 180^\circ = 720^\circ \]
  2. Isolate the term \((n-2)\): Divide both sides of the equation by \(180^\circ\). \[ n-2 = \frac{720^\circ}{180^\circ} \]
  3. Calculate the value: Perform the division. \[ n-2 = 4 \]
  4. Solve for n: Add 2 to both sides of the equation to find the number of sides. \[ n = 4 + 2 \] \[ n = 6 \]

The calculation shows that the polygon has 6 sides.

Identifying the Polygon Type

A polygon with 6 sides is known as a Hexagon. Therefore, the regular polygon described in the question is a Hexagon.

Comparing this result with the given options:

  • Hexagon (6 sides)
  • Octagon (8 sides)
  • Nonagon (9 sides)
  • Decagon (10 sides)

Our calculated value of \(n=6\) corresponds to a Hexagon.

Was this answer helpful?

Similar Questions

  1. If the quadrilateral has an inscribed circle, then the sum of a pair of opposite sides equals:

  2. ABCD is a cyclic quadrilateral. The bisectors of the angles A, B, C and D cut the circle at P, Q, R and S respectively. What is ∠PQR + ∠RSP equal to?

  3. What is the magnitude (in radian) of the interior angle of a regular pentagon?

  4. OABC is a rhombus whose three vertices lie on a circle with centre at O. If the area of the rhombus is 32√3 square cm, then what is the radius of the circle ?

  5. ABCD is a square field with AB = x. A vertical pole OP of height 2x stands at the centre O of the square field. If ∠APO = θ, then what is cot θ equal to ?

  6. A square is inscribed in a right angled triangle with legs p and q and has a common right angle with triangle. The diagonal of the square is given by

  7. ABCDA is a con-cyclic quadrilateral of a circle ABCD with radius r and centre at O. If AB is the diameter and CD is parallel and half of AB and if the circle completes one rotation about the centre O, then the locus of the middle point of CD is a circle of radius:

  8. The diagonals of a rhombus are of length 20 cm and 48 cm. What is the length of a side of the rhombus?

  9. The area of a regular hexagon of side ‘a’ is equal to

  10. Two parallel sides of a trapezium are 29 cm and 21 cm. Non-parallel sides are equal and each is of length 8.5 cm. What is the area of the trapezium?


Important Questions from Quadrilaterals

  1. The ratio between the length and breadth of a rectangular park is 3 : 2. If a man cycling along the boundary at the speed of 12 km per hour completes one round in 8 minutes, then the area of the park in square meter will be

  2. The side of a rhombus is 26 cm. The length of one of its diagonals is 20 cm. The sum of the lengths of the diagonals of this rhombus is equal to the perimeter of a rectangle. If the difference between the length and breadth of the rectangle is 6 cm, then what is the area of the rectangle?

  3. PQRS is a cyclic quadrilateral. If ∠P is 4 times ∠R, and ∠S is 3 times ∠Q, then the average of ∠Q and ∠R is:

  4. ABCD is a trapezium in which AB || DC and DC is perpendicular to BC. If ∠DAB = 110°, then ∠ABC - ∠ADC =_____.

  5. If the quadrilateral has an inscribed circle, then the sum of a pair of opposite sides equals:

Need Expert Advice?
Test Series
CDS img
Defence
UPSC CDS 2026 Mock Test Series
536 Tests 4 Tests Free
1713 Attempts
4.3(174)
English, Hindi
More Questions from CDS

Start Your Preparation with Prepp Mobile App

Download the app from Google Play & App Store
Download the app from Google Play & App Store
Prepp Mobile App