Find the area of the intersection between two circles
Geometry
Level
3
Find the area of intersection of two circles as shown in the figure above. Round of to 1 decimal.
The answer is 7.6.
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1 solution
Equation of the two circles x x + y y = 25 -- C1
Circle C2 (x-7) (x-7) + y y = 16
x x - 14x + 49 +y y = 16 x x - 14x +y y = - 33
Solving for intersection points A and B of the two circles
14x - 49 =9 14x = 58 x = 4.14 y = +/- 2.79 Co-ords of A are (4.14, 2.79) and of B are (4.14 ,-2.79)
Length of the chord AB = 5.58
Angle subtended by the chord AC1B = 2 * invsin(2.79/5) = 67.84 degrees
Area of the sector C1AB = (67.84/360) 25 22/7 = 14.8 sqcm Area of the triangle C1AB = 11.58
Area of the segment 1 =14.8 - 11.58 = 3.22 sqcm
Angle subtended the chord on circle 2 AC2B = 2 * 44.2267712 = 88.44 degrees Area of the sector AC2B = (88.44/360) 16 22/7 = 12.35 sqcm Area of the triangle AC2B = 7.999 sqcm Area of the segment 2 = 12.35 - 7.999 = 4.35 sqcm
Total intersected area = Area of Segment 1 + Area of Segment 2 = 7.57 sqcm