RRB ALP Maths Mensuration (Polygons): Formulas, Properties & Questions

A polygon is a closed two-dimensional figure made up of three or more straight line segments. In the RRB ALP exam, polygon-based questions generally test concepts such as interior and exterior angles, diagonals, perimeter, and formulas related to regular polygons. 

A clear understanding of these concepts helps you solve geometry and mensuration questions more accurately and improve your overall score in the RRB ALP exam. 

Polygons in Mensuration

Mensuration of polygons primarily deals with calculating their area and perimeter. While geometry focuses on angles, sides, and diagonals, this session emphasises the quantitative measurement aspects. 

Our main focus will be on understanding the properties and formulas of regular hexagons, crucial for various competitive examinations.

Regular Hexagon: Properties and Formulas

A regular hexagon (समषट्भुज / खटभुज) is a polygon characterised by six equal sides and six equal internal angles.

1. Internal and External Angles:

• The internal angle of a regular hexagon is 120°.

• The external angle of a regular hexagon is 60°.

• The external angle for any regular polygon is calculated as 360° / n, where 'n' is the number of sides. For a hexagon, n=6, so 360°/6 = 60°.

• The internal angle is then 180° - External angle = 180° - 60° = 120°.

2. Decomposition into Equilateral Triangles:

• A regular hexagon can be effectively divided into six equilateral triangles by connecting its centre to each of its vertices.

• Each of these triangles has angles of 60°, 60°, 60°, confirming their equilateral nature.

3. Perimeter of a Regular Hexagon:

• The perimeter of a regular hexagon is the total length of its boundary, which is the sum of its six equal sides.

• Formula: Perimeter = 6a, where 'a' represents the length of one side.

4. Area of a Regular Hexagon:

• The area of a regular hexagon is a frequently tested concept.

• Formula (Derivation): The area can be derived by summing the areas of the six equilateral triangles it contains. (Memory Tip: This approach helps in recalling the formula even if forgotten.)

• Area of one equilateral triangle = (√3 / 4) * a²

• Area of Hexagon = 6 × (Area of one equilateral triangle)

• Area of Hexagon = 6 × (√3 / 4) * a²

• Simplified Formula: Area of Hexagon = (3√3 / 2) * a²

Regular Hexagon: Inscribed and Circumscribed Circles

Understanding the relationship between circles and regular hexagons is crucial for solving many problems.

1. Circumradius (Radius of the Circumscribed Circle):

• When a circle circumscribes a regular hexagon (meaning the hexagon is inscribed in the circle), the circumradius (R) of the hexagon is equal to its side length (a).

• R = a

2. Inradius (Radius of the Inscribed Circle):

• When a circle is inscribed within a regular hexagon (meaning the hexagon circumscribes the circle), the inradius (r) is the perpendicular distance from the centre to any side. This is also known as the apothem.

• This distance corresponds to the height of one of the equilateral triangles formed within the hexagon.

• Formula: r = (√3 / 2) * a

3. Ratio of Circumradius to Inradius (R:r):

• This ratio is frequently asked in competitive examinations. 

• R : r = a : (√3 / 2) * a

• R : r = 1 : (√3 / 2)

• R : r = 2 : √3

Comparative Structures & Exam Relevance

The study of polygons in mensuration, particularly regular hexagons, is highly important for various competitive exams including SSC CGL, CHSL, NTPC, Group D, and RRB ALP.

 While basic shapes like triangles and quadrilaterals are common, questions on more complex polygons are increasingly frequent. It is important to understand these concepts and remember the formulas for exam preparation. 

Solved Examples

Here are some solved examples demonstrating the application of regular hexagon formulas:

Problem 1: Perimeter of a Regular Hexagon

• Question: The perimeter of a regular hexagon is 138 cm. How long is each side?

• Solution:

• Perimeter = 6a

• 138 = 6a

• a = 23 cm

Problem 2: Area of a Regular Hexagon from Perimeter

• Question: A regular hexagon has a perimeter of 84 cm. What is its area?

• Solution:

• Perimeter = 6a => 84 = 6a => a = 14 cm

• Area = (3√3 / 2) * a²

• Area = (3√3 / 2) * 14² = (3√3 / 2) * 196

• Area = 3√3 * 98 = 294√3 cm²

Problem 3: Perimeter of a Hexagon from a Diagonal

• Question: ABCDEF is a regular hexagon. If BE = 14 cm, what is the perimeter of the hexagon?

• Solution:

• In a regular hexagon, the longest diagonal (connecting opposite vertices, like BE) is twice the side length (2a).

• BE = 2a

• 14 = 2a => a = 7 cm

• Perimeter = 6a = 6 * 7 = 42 cm

Problem 4: Scaling of Area in a Regular Hexagon

• Question: If the side of a regular hexagon is tripled, then how does its area change?

• Solution:

• Area Formula: Area ∝ a²

• [Key Principle]: When side 'a' is tripled (becomes 3a), then a² becomes (3a)² = 9a².

• Therefore, the area increases 9 times (9-fold).

Problem 5: Area of a Hexagon Inscribed in a Circle

• Question: A regular hexagon is inscribed inside a circle of radius 18 cm. Find the area of the hexagon.

• Solution:

• When a regular hexagon is inscribed in a circle, the radius of the circle (R) is equal to the side length (a) of the hexagon.

• Given R = 18 cm, so a = 18 cm.

• Area = (3√3 / 2) * a²

• Area = (3√3 / 2) * 18² = (3√3 / 2) * 324

• Area = 3√3 * 162 = 486√3 cm²

Problem 6: Ratio of Area of Regular Hexagon to Square with Same Side

• Question: A regular hexagon and a square have the same side. What is the ratio of their areas?

• Solution:

• Let the side be 'a'.

• Area of Regular Hexagon = (3√3 / 2) * a²

• Area of Square = a²

• Ratio (Hexagon Area : Square Area) = [(3√3 / 2) * a²] : [a²]

• Ratio = 3√3 : 2