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ICSE-VIII-Mathematics

16: Understanding Shapes (Including Polygons) Class 8 Maths

with Solutions - page 2

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  • #4-ii
    2340°
    Ans : Let no. of sides = n
    Sum of angles = 2340°
    (n - 2) × 180° = 2340°
    ⇒ n - 2 = 2340/180
    ⇒ n - 2 = 13
    ⇒ n = 13 + 2 = 15
    Which is a whole number.
    Hence it is possible to have a polygon, the sum of whose interior angles is 2340°.
  • #4-iii
    7 right-angles
    Ans : Let no. of sides = n
    Sum of angles = 7 right angles = 7 ×90 = 630°
    (n - 2) × 180° = 630°
    ⇒ n - 2 = 630/180
    ⇒ n - 2 = 7/2
    ⇒ n = 7/2 + 2
    ⇒ n = 11/2
    Which is not a whole number. Hence it is not possible to have a polygon, the sum of whose interior angles is 7 right-angles.
  • #4-iv
    4500°
    Ans : Let no. of sides = n
    (n - 2)×180° = 4500°
    ⇒ n - 2 = 4500/180
    ⇒ n - 2 = 25
    ⇒ n = 25 + 2
    ⇒ n = 27
    Which is a whole number.
    Hence it is possible to have a polygon, the sum of whose interior angles is 4500°.
  • #5
  • #5-i
    If all the angles of a hexagon are equal; find the measure of each angle.
    Ans : No. of sides of hexagon, n = 6
    Let each angle be = x°
    Sum of angles = 6x°
    (n - 2)× 180° = Sum of angles
    ⇒ (6 - 2)× 180° = 6x°
    ⇒ 4× 180 = 6x
    ⇒ x = (4 × 180)/6
    ⇒ x = 120˚
    ∴ Each angle of hexagon = 120˚
  • #5-ii
    If all the angles of a 14-sided figure are equal; find the measure of each angle.
    Ans : No. of sides of polygon, n = 14
    Let each angle = xËš
    ∴ Sum of angles = 14x˚
    ∴ (n - 2) × 180˚ = Sum of angles of polygon
    ∴ (14 - 2) × 180˚ = 14x
    12 × 180˚ = 14x
    ⇒ x = (12 × 180)/14
    ⇒ x = 1080/7
    ⇒ x = (154 2/7)˚
  • Qstn #6
    Find the sum of exterior angles obtained on producing, in order, the sides of a polygon with :
  • #6-i
    7 sides
    Ans : No. of sides n = 7
    Sum of interior & exterior angles at one vertex = 180°
    Sum of interior & exterior angles = 7 × 180˚
    = 1260Ëš
    Sum of interior angles = (n - 2) × 180˚
    = (7 - 2) × 180˚
    = 900Ëš
    ∴ Sum of exterior angles = 1260˚ - 900˚
    = 360Ëš
  • #6-ii
    10 sides
    Ans : No. of sides n = 10
    Sum of interior and exterior angles = 10 × 180˚
    = 1800Ëš
    But sum of interior angles = (n - 2) × 180˚
    = (10 - 2) × 180˚
    = 1440Ëš
    ∴ Sum of exterior angles = 1800 - 1440
    = 360Ëš
  • #6-iii
    250 sides.
    Ans : No. of sides n = 250
    Sum of all interior and exterior angles = 250 × 180˚
    = 45000Ëš
    But sum of interior angles = (n - 2) × 180˚
    = (250 - 2) × 180˚
    = 248 × 180˚
    = 44640Ëš
    ∴ Sum of exterior angles = 45000 - 44640
    = 360Ëš
  • Qstn #7
    The sides of a hexagon are produced in order. If the measures of exterior angles so obtained are (6x - 1)°, (10x + 2)°, (8x + 2)° (9x - 3)°, (5x + 4)° and (12x + 6)° ; find each exterior angle.
    Ans : Sum of exterior angles of hexagon formed by producing sides of order = 360°
    ∴ (6x - 1)˚ - (10x + 2)˚ + (8x + 2)˚ + (9x - 3)˚ + (5x + 4)˚ + (12x + 6)˚ = 360˚
    50x + 10Ëš = 360Ëš
    ⇒ 50x = 360˚ - 10˚
    ⇒ 50x = 350˚
    ⇒ x = 350/70
    ⇒ x = 7
    ∴ Angles are (6x - 1)˚ : (10x + 2)˚ : (8x + 2)˚ : (9x - 3)˚ : (5x + 4)˚ and (12x + 6)˚
    i.e. (6 × 7 - 1)˚ : (10 × 7 + 2)˚ : (8 × 7 + 2)˚ : (9 × 7 - 3)˚ : (5 × 7 + 4)˚ : (12 × 7 + 6)˚
    i.e. 41° ; 72°, 58° ; 60° ; 39° and 90°
  • Qstn #8
    The interior angles of a pentagon are in the ratio 4: 5: 6: 7: 5. Find each angle of the pentagon.
    Ans : Let the interior angles of the pentagon be 4x, 5x, 6x, 7x, 5x.
    Their sum = 4x + 5x + 6x + 7x + 5x = 21x
    Sum of interior angles of a polygon = (n - 2) × 180˚
    = (5 - 2) × 180˚
    = 540Ëš
    ∴ 27x = 540
    ⇒ x = 540/27
    ⇒ x = 20˚
    ∴ Angles are 4 × 20˚ = 80˚
    5 × 20˚ = 100˚
    6 × 20˚ = 120˚
    7 × 20˚ = 140˚
    5 × 20˚ = 100˚
  • Qstn #9
    Two angles of a hexagon are 120° and 160°. If the remaining four angles are equal, find each equal angle.
    Ans : Two angles of a hexagon are 120°, 160°
    Let remaining four angles be x, x, x and x.
    Their sum = 4x + 280°
    But sum of all the interior angles of a hexagon
    = (6 - 2) × 180˚
    = 4 × 180˚
    = 720Ëš
    ∴ 4x + 280˚ = 720˚
    ⇒ 4x = 720˚ - 280˚⇒ 4x = 440˚
    ⇒ x = 110˚
    ∴ Equal angles are 110˚ (each)
  • Qstn #10
    The figure, given below, shows a pentagon ABCDE with sides AB and ED parallel to each other, and ∠B: ∠C : ∠D = 5: 6: 7.
  • #10-i
    Using formula, find the sum of interior angles of the pentagon.
    Ans : Sum of interior angles of the pentagon
    = (5 - 2) × 180˚
    = 3 × 180˚ = 540˚ [∵ sum for a polygon of x sides = (x - 2) × 180˚]
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