Geometry is one of the most important and easiest topics of Quantitative Aptitude, if you prepare it well. Like any other topic, geometry too needs a lot of practice. In this article, we will discuss about polygons and its important properties. The important properties and formulas discussed in this article find direct application in questions. So, make sure you go through this article in detail.
Polygon: A polygon is a closed plane figure formed by three or more line segments, called the sides of the polygon. Each side intersects exactly two other sides at their endpoints. The points of intersection of the sides are known as vertices. The term "polygon" is used to mean a convex polygon, that is, a polygon in which each interior angle has a measure of less than 180°. The following figures are polygons:
The following figures are not polygons:
Number of sides | Name of the Polygon |
---|---|
3 | Triangle |
4 | Quadrilateral |
5 | Pentagon |
6 | Hexagon |
7 | Heptagon |
8 | Octagon |
9 | Nonagon |
10 | Decagon |
Some Basic Terminologies
Convex polygon: A polygon in which none of the interior angles is more than 180° is called a convex polygon.
Concave polygon: A polygon in which at least one of the interior angles is more than 180° is called a concave polygon.
Regular polygon: A polygon which has all its sides and angles equal is called a regular polygon.
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.
Solved Examples
Example: What is the area of a regular octagon of side 8 cm?
Solution: The area of the octagon can be expressed as the difference of the larger square that comprises the octagon and the 4 right triangles at the corners that you can deduct.
Each of the right triangles will have the two perpendicular sides as 8/ √2.
The side of the bigger square will be 8 + 8/√2 + 8/√2 = (1+ √2) 8.
Squaring both sides, we get the area of the square as (1 + 2 + 2√2)8^{2}.
The sum of areas of the four triangles = 4 * ½ *(1/√2)^{2} * 8^{2}.
The difference between the two works out to be (2+2√2) * 8^{2}, that will be area of the octagon.
Hexagon
Here, a = side of the hexagon;
d= diameter of the circle inscribed (distance between two parallel sides);
p= perimeter of the hexagon.
The hexagon can be divided into 6 equilateral triangles with side=a
Hence, area of hexagon=
Area of hexagon=
Length of the diagonals=
Let us solve some questions based on the above discussed concepts.
Solved Examples
Example 1: 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)
Solution: Answer- Option 3
We know that the lower limiting perimeter of any polygon S^{1} is the circumference of the inscribed circle (2*π).
The upper limiting perimeter of any polygon S2 is the circumference of the circumscribed circle: 2*π
This difference of perimeter reduces as the number of sides increase.
Breaking up the expression into L1(13)/L2(17) + 2*π/L2(17).
Both the individual terms will be very close to 1, but greater than one.
Hence, option 3 is the answer.
Question 2: 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)
Solution: Answer- option 1
Let third side be x.
S = (20 + 10 + x)/2 = (30 + x)/2.
Now use Hero's formula for the area of the triangle and we can find the answer as 1^{st} option.
Question 3: What is the area of a regular octagon of side a?
Solution: Answer- option 1
The area of the octagon can be expressed as the difference of the larger square that comprises the octagon and the 4 right triangles at the corners that we can deduct.
Each of the right triangles will have the two perpendicular sides as a / √2.
The side of the bigger square will be a + a/√2 + a/√2 = (1 + √2) a.
On squaring we get the area of the square as (1 + 2 + 2√2)a^{2} and that of 4 triangles as 4 x ½ x(1/√2)^{2} x a^{2}.
The difference between the two equals (2+2√2) x a^{2}.
Question 4: ABCDEF is a regular hexagon AP: PB = 5: 2 and PR is parallel to AF. Find the ratio of lengths of AF to PR.
Solution: Answer- Option 3
Here, ΔAQF is an equilateral triangle
So AF = AQ
But AF || PR. Then Δ AQF and Δ PQR are similar
So
Key Learning
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