A circle of radius r is drawn with its centre on the circumference of another circle of radius r.Show that the common area to both circles is 2r^2 (Π/3-√3/4)
Answers
Given : A circle of radius r is drawn with its centre on the circumference of another circle of radius r.
To Find : Show that the common area to both circles is 2r² (π/3-√3/4)
Solution:
Refer the Figure attached
common area to both circles = Area of 2 sectors of circle - Quadrilateral ACBD formed in between
Each Sector angle = 120° ( 60° + 60° = 120°) ( angles of Equilateral Triangle)
Quadrilateral can be divided in 2 Equilateral Triangle with side r
Area of a sector = (120°/360°)πr² = πr²/3
Area of a triangle = (√3 / 4)r²
common area to both circles = Area of 2 sectors of circle - Area of 2 Equilateral Triangles
= 2 ( area of a sector - area of a equilateral triangle)
= 2 ( πr²/3 - (√3 / 4)r² )
= 2r² ( π/3 - √3 / 4)
QED
Hence Shown
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