Area of a Polygon



In area of a polygon we will learn about polygon, regular polygon, central point of the polygon, radius of the inscribed circle of the polygon, radius of the circumscribed circle of a polygon and solved problems on area of a polygon.



Polygon: A figure bounded by four or more straight lines is called a
              polygon.


Regular Polygon: A polygon is said to be regular when all its sides are
                         equal and all its angles are equal.

A polygon is named according to the number of sides it contains.

Given below are the names of some polygons and the number of sides contained by them.

  • Quadrilateral - 4
  • Pentagon - 5
  • Hexagon - 6
  • Heptagon - 7
  • Octagon - 8
  • Nonagon - 9
  • Decagon - 10
  • Undecagon - 11
  • Dodecagon - 12
  • Quindecagon -15


    Central Point of a Polygon:

    The inscribed and the circumscribed circles of a polygon have the same centre, called the central point of the polygon.

    Radius of the Inscribed Circle of a Polygon:

    The length of perpendicular from the central point of a polygon upon any one of its sides, is the radius of the inscribed circle of the polygon.
    The radius of the inscribed circle of a polygon is denoted by r.

    Radius of the Circumscribed Circle of a Polygon:

    The line segment joining the central point of a polygon to any vertex is the radius of the circumscribed circle of the polygon. The radius of the circumscribed circle of a polygon is denoted by R.

    In the figure given below, ABCDEF is a polygon having central point O and one of its sides a unit. OL ⊥ AB.
    Then, OL = r and OB = R



    Area of a polygon of n sides

            = n × (area ∆OAB) = n × 1/2 × AB × OL

            = (n/2 × a × r)

    Now, A = 1/2 nar ⇔ a = 2A/nr ⇔ na = 2A/r ⇔ Perimeter = 2A/r

    From right ∆OLB, we have:

    OL2 = OB2 - LB2 ⇔ r2 = {R2 - (a/2)2}

                          ⇔ r = √(R2 - (a2/4)

    Therefore, area of the polygon = {n/2 × a × √(R2 - a2/4) square units.





    In area of a polygon some of the particular cases such as;

    (i) Hexagon:

        OL2 = (OB2 - LB2)

            = {a2 - (a/2)2} = (a2 - a2/4) = 3a2/4

    ⇒ OL = {(√3)/2 × a}

    ⇒ Area ∆OAB = 1/2 × AB × OL

          = {1/2 × a × (√3)/2 × a} = (√3)a2/4

    ⇔ area of hexagon ABCDEF = {6 × (√3)a2/4} square units

                = {3(√3)a2/2} square units.

    Therefore, area of a hexagon = {3(√3)a2/2} square units.


    (ii) Octagon:

        BM is the side of a square whose diagonal is BC = a.

    Therefore, BM = a/√2.

    Now, OL = ON + LN

        = ON + BM = (a/2 + a/√2)

    ⇔ Area of given octagon

        = 8 × area of ∆OAB = 8 × 1/2 × AB × OL

        = 4 × a × (a/2 + a/√2) = 2a2 (1 + √2) square units.

    Therefore, area of an octagon = 2a2 (1 + √2) square units.


    We will solve the examples on different names of the area of a polygon.

    Area of a Polygon



    1. Find the area of a regular hexagon each of whose sides measures 6 cm.

    Solution:

    Side of the given hexagon = 6 cm.

    Area of the hexagon = {3√(3)a2/2} cm2

            = (3 × 1.732 × 6 × 6)/2 cm2

            = 93.528 cm2.


    2. Find the area of a regular octagon each of whose sides measures 5 cm.

    Solution:

    Side of the given octagon = 5 cm.

    Area of the octagon = [2a2 (1 + √2) square units

            = [2 × 5 × 5 × (1 + 1.414)] cm2

            = (50 × 2.414) cm2

            = 120.7 cm2.


    3. Find the area of a regular pentagon each of whose sides measures 5 cm and the radius of the inscribed circle is 3.5 cm.

    Solution:

    Here a = 5 cm, r = 3.5 cm and n = 5.

    Area of the pentagon = (n/2 × a × r) square units

            = (5/2 × 5 × 7/2) cm2         = 43.75 cm2.


    4. Each side of a regular pentagon measures 8 cm and the radius of its circumscribed circle is 7 cm. Find the area of the pentagon.

    Solution:

    Area of the pentagon = {n/2 × a × √(R2 - a2/4) square units

            = {5/2 × 8 × √(72 - 64/4)} cm2

            = {20 × √(49 - 16)} cm2         = (20 × √33) cm2

            = (20 × 5.74) cm2         = (114.8) cm2.

    Related Links:



    Area of a Trapezium

  • Area of a Trapezium
  • Area of a Polygon

  • Area of a Trapezium - Worksheet
  • Worksheet on Trapezium
  • Worksheet on Area of a Polygon

  • 8th Grade Math Practice

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