Concept of Polygons

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In this article we will cover the questions based on the concept of Polygons. Once you are through with the concept of Polygons, then you will be able to solve all the questions based on this topic. Problems on Polygons are important from Geometry point of view.

Polygon: A polygon is a 'n'sided closed figure where n ≥ 3. A polygon with
- 3 sides is called a triangle.
- 4 sides is called a quadrilateral.
- 5 sides is called a pentagon.
- 6 sides is called a hexagon.
- 7 sides is called a heptagon.
- 8 sides is called a octagon.
- 9 sides is called a nonagon.
- 10 sides is called a decagon.

Each side intersects exactly two other sides at their endpoints.
The points of intersection of the sides are known as vertices of the polygon.
The term 'polygon' refers to a convex polygon, that is, a polygon in which each interior angle has a measure of less than 180°.

A concave polygon is a polygon in which atleast one interior angle is more than 180°.

Regular Polygon is a symmetrical polygon. All sides are equal, all angles are equal. All regular polygons are convex.

Perimeter of a polygon: The perimeter of a polygon is the sum of the lengths of all its sides.

Area of a polygon: The region enclosed within a figure is called its area.

Diagonal of a polygon: The segment joining any two non-consecutive vertices is called a diagonal.

Properties of Polygons:

A 'n' sided polygon has 'n' interior angles and 'n' exterior angles.
1) Number of diagonals = n(n - 3)/2.
2) Sum of interior angles = (n - 2)180°.
3) Sum of exterior angles = 360° (For convex polygons)

For a regular polygon:

1) Each interior angle = i such that n × i = (n - 2)180°
2) Each exterior angle = e such that n × e = 360°
3) i + e = 180°
4) Area of a regular polygon =

Regular Hexagon:

It can be divided into 6 equilateral triangles of side 'a'.
Area of the equilateral triangle =

In hexagon ABCDEF of side 'a'
1) AB = BC = CD = DE = EF = FA = a
2) AC = BD = CE = DF = EA = FB =

a
3) AD = BE = CF = 2a

Now let's consider some questions to understand the concept.

Read More: Concept of Escalator

Q1) Sum of all the interior angles of a regular polygon is 3240°. What is the measure of it's exterior angle?
Solution:
(n - 2)180° = 3240° i.e. n = 20
20n = 360° i.e. e = 18°

Q2) The perimeter of a square, a regular octagon and a hexagon are equal. If their areas are denoted by S, O and H respectively, then which of the following is true.
1) S > O > H
2) O > S > H
3) H > S > O
4) O > H > S
Solution:
When perimeter is same, then more are the number of sides more is the area. Hence, option 4.

Q3) How many regular polygons with number of sides 'n' exist such that all the angles (in degrees) of the polygon are integers?
Solution:
Each interior angle of a regular polygon is 180 - 360/n.
Total factors of 360 = 24
Regular Polygons satisfying the conditions specified above are 22.

Q4) Find the area of an octagon (which is not a regular polygon) whose perimeter is 20 cm and which circumscribes a circle of radius 3 cm.
Solution:
Area of the polygon = inradius × semi-perimeter = 3 × 20/2 = 30 cm²

Q5) In hexagon ABCDEF, P, Q, R are the mid points of sides DE, FA, BC. What is the ratio of the area of Δ PQR to the area of the hexagon.
Solution:
CF = 2a, AB = a, RQ = (AB + CF)/2 = (a + 2a)/2 = 1.5a
Required ratio = 3/8.

Q6) A square of side 's' units is cut down into a regular octagon. What is the side of the regular octagon?
Solution:
Let us cut a right angled triangle from the 4 corners of the square to form an octagon. Let 'x' be the side of the right angled triangle, such that the hypotnuse of the right angled triangle be the side of the octagon.

Questions on polygons from Previous year CAT papers:

Q1) There is a circle of radius 1 cm. Each member of a sequence of regular polygons S1(n), n = 4, 5, 6, ... where n = number of sides of the polygon, is circumscribing the circle and each member of the sequence of regular polygons S2(n), n = 4, 5, 6, ...., where n is the number of sides of the polygon, is inscribed in the circle. Let L1(n) and L2(n) denote perimeters of the corresponding polygons of S1(n) and S2(n). Then is (CAT 1999)

1) Greater than π/4 but less than 1
2) Greater than 1 and less than 2
3) Greater than 2
4) Less then π/4

Answer: Option 3.

Q2) Euclid has just created a triangle whose longest side is 20. If the length of the other side is 10 cm and the area of the triangle is 80 sq. cm, then what is the length of the third side? (CAT 2001)

1) √260
2) √250
3) √240
4) √270

Answer: Option 1.

Q3) Let ABCDEF be a regular hexagon. What is the ratio of the area of Δ ACE to that of the hexagon ABCDEF? (CAT 2015 Type)

1) 1/3
2) 1/2
3) 2/3
4) 5/6

Answer: Option 2.