A regular hexagon is inscribed in a circle of radius r. The perimeter of the regular hexagon is
(a) 3r
(b) 6r
(c) 9r
(d) 12r
Solution:
The diagram showing the inscribed hexagon is shown below:
Since it is a regular hexagon the internal angles of the polygon are 120° each. OA and OB are also the radius of the circle in which the hexagon is inscribed. OA and OB also bisect the angles ∠OAF and ∠OBC respectively. Therefore
∠OAB = 60°
∠OBA = 60°
Since the two base angles are 60°, ∠AOB = 60°
△AOB is therefore an equilateral triangle, therefore
AB = r
Since it is a regular hexagon,
AB = BC = CD = DE = EF = FA = r
Hence the perimeter of the hexagon is = 6 × r = 6r
The correct choice is (b).
✦ Try This: A regular hexagon is inscribed in a circle of radius r. The area of the regular hexagon is (a) (√3/2)r² (b) (3√3/4)r² (c) (3√3/2)r² (d) 3√3r²
The diagram showing the inscribed hexagon is shown below:
Since it is a regular hexagon the internal angles of the polygon are 120° each. OA and OB are also the radius the circle in which the hexagon is inscribed. OA and OB also bisect the angles ∠OAF and ∠OBC respectively. Therefore
∠OAB = 60°
∠OBA = 60°
Since the two base angles are 60°, ∠AOB = 60°
△AOB is therefore an equilateral triangle, therefore
AB = r
Since it is a regular hexagon,
AB = BC = CD = DE = EF = FA = r
Also all triangles i.e.
△AOB = △BOC = △COD = △DOE = △EOF = △FOA = equilateral triangles with side r
The area of the hexagon can be calculated by calculating the area of any one of these triangles and multiplying by 6 to find the area of the hexagon.
Area of a triangle = √s(s - a)(s - b)(s - c)
Where s = (a + b + c)/2 = semi perimeter of the triangle
a, b and c are sides of the triangle
In the given problem a = b = c = r and s = 3r/2
Area of the triangle = √(3r/2)(3r/2 - r)(3r/2 - r)(3r/2 -r)
Area of the triangle = √(3r/2)(r/2)(r/2)(r/2)
Area of Triangle = √(3/16)r⁴
Area of Triangle = (r²/4)√3
Area of Hexagon = 6 × (r²/4)√3
Area of Hexagon = (3√3/2)r²
The correct choice is c)
☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 11
NCERT Exemplar Class 8 Maths Chapter 11 Problem 9
A regular hexagon is inscribed in a circle of radius r. The perimeter of the regular hexagon is (a) 3r (b) 6r (c) 9r (d) 12r
Summary:
A regular hexagon is inscribed in a circle of radius r. The perimeter of the regular hexagon is 6r
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