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Two circles, each of radii 2 cm, intersect each other such that the center of each one passes through the center of the other. What is the area (in sq cm) of the intersecting region?
- a)2π / 3 - √3
- b)8π / 3 - 2√3
- c)2π / 3 - √3 / 2
- d)π / 3 - 2√3
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Two circles, each of radii 2 cm, intersect each other such that the ce...
Let the centers of the two circles be B and C, as shown below.
Required area = A(region ABCD).
Because both circles have equal radii, AB = AC = BC = BD = CD = 2 cm Hence, ΔABC and ABCD are equilateral triangles.
Area (region ABCD) = 2[A(ΔABC)] + 4(Area of minor segment AB)
A(ΔABC) = (√3 / 4) x (2)2 = √3 sq. cm
Area of minor segment AB = Area of minor sector BAC - Area of ΔABC
∴ A(minor segment AB) = (1 / 6) x π x 22 - √3 = 2π / 3 - √3
∴ Total area = 2 x (√3) + 4 x (2π / 3 - √3) = (8π / 3 - 2√3)
Hence, option 2.
Because both circles have equal radii, AB = AC = BC = BD = CD = 2 cm Hence, ΔABC and ABCD are equilateral triangles.
Area (region ABCD) = 2[A(ΔABC)] + 4(Area of minor segment AB)
A(ΔABC) = (√3 / 4) x (2)2 = √3 sq. cm
Area of minor segment AB = Area of minor sector BAC - Area of ΔABC
∴ A(minor segment AB) = (1 / 6) x π x 22 - √3 = 2π / 3 - √3
∴ Total area = 2 x (√3) + 4 x (2π / 3 - √3) = (8π / 3 - 2√3)
Hence, option 2.
Most Upvoted Answer
Two circles, each of radii 2 cm, intersect each other such that the ce...
Since the radii of the circles are both 2 cm, they have equal length. The distance between the centers of the circles is also 2 cm, so the centers of the circles are at the endpoints of a diameter of each circle. Thus, the circles are tangent at their centers. The area of the intersecting region is just the area of a circle with radius 2 cm, which is $\pi(2^2)=\boxed{4\pi}$.
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Two circles, each of radii 2 cm, intersect each other such that the center of each one passes through the center of the other. What is the area (in sq cm) of the intersecting region?a)2π/ 3 -√3b)8π/ 3 - 2√3c)2π/ 3 -√3/ 2d)π/ 3 - 2√3Correct answer is option 'B'. Can you explain this answer?
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