Question

If the area of the red rectangle is 27 cm2 and O is the centre of a bigger circle, what is the total area (in cm) covered by both circles? о 24 cm​

Answer 1

Step-by-step explanation:

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Answer 2

• The total area covered by both circles is equal to 117π cm² or 367.38 cm² .

Given :-

• Area of red rectangle = 27 cm²
• O is the centre of bigger circle .
• One side of outer rectangle is equal to 24 cm .

To Find :-

• Total area of both circles ?

Formula used :-

• Area of rectangle = Length × Breadth .
• Area of circle = π•r² { r = radius }
• π = 3.14
• a² + b² = (a + b)² - 2ab

Construction :- { Refer to image }

• O' = Centre of smaller circle .
• r = Radius of smaller circle .
• R = Radius of bigger circle .
• ABCD red rectangle .
• G = Point of intersection of both circles at side AD .

Solution :-

from image we can see that,

→ HO' = O'G = r { Radius of smaller circle }

→ GO = OI = R { Radius of bigger circle }

As we can see that,

→ HI = 24 cm { Parallel and equal to to given side of outer rectangle }

→ HO' + O'G + GO + OI = 24

→ r + r + R + R = 24

→ 2r + 2R = 24

→ 2(r + R) = 24

→ (r + R) = 12 ------------- Equation (1)

Now,

→ Area of red rectangle (ABCD) = 27 cm² { given }

→ Length × Breadth = 27

→ AB × BC = 27

as we can see that, AB = GO = R and BO = OC = EO' = O'F = r

→ R × (r + r) = 27

→ R × 2r = 27

→ 2rR = 27 -------------- Equation (2)

Now,

→ Area of smaller circle + Area of bigger circle

→ π•r² + π•R²

→ π(r² + R²)

using a² + b² = (a + b)² - 2ab,

→ π[(r + R)² - 2rR]

putting value of Equation (1) and Equation (2) we get,

→ π[(12)² - 27]

→ π(144 - 27)

→ 117π

finally putting value of π as 3.14,

→ 117 × 3.14

→ 367.38 cm² (Ans.)

Hence, the total area covered by both circles is equal to 367.38 cm² .

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