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A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?
- a)2 : 3
- b)3 : 4
- c)1 : 4
- d)1 : 2
- e)None of these
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A circle is inscribed in a given square and another circle is circums...
Let the side of a square be a units
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Radius of the inscribed circle = a/2
Radius of the circle circumscribing the square = a/√2
Hence, ratio of the areas = (a/2)2 : (a/√2)2 = 1:2
Most Upvoted Answer
A circle is inscribed in a given square and another circle is circums...
The given problem involves a square and two circles - one that is inscribed in the square and another that circumscribes the square. We are asked to find the ratio of the area of the inscribed circle to that of the circumscribed circle.
Let's denote the side length of the square as "s". Since the square is inscribed in a circle, the diagonal of the square will be equal to the diameter of the circumscribed circle. The diagonal of the square can be found using the Pythagorean theorem:
Diagonal of square = √(s^2 + s^2) = √2s
Area of the square = s^2
- Inscribed Circle:
The diameter of the inscribed circle is equal to the side length of the square. Therefore, the radius of the inscribed circle is half the side length of the square, or s/2.
Area of the inscribed circle = π(s/2)^2 = πs^2/4
- Circumscribed Circle:
The diameter of the circumscribed circle is equal to the diagonal of the square, which we found to be √2s. Therefore, the radius of the circumscribed circle is half the diagonal of the square, or (√2s)/2 = √2s/2.
Area of the circumscribed circle = π(√2s/2)^2 = π(2s/4) = πs^2/2
- Ratio of Areas:
Now we can find the ratio of the area of the inscribed circle to that of the circumscribed circle:
Ratio = (Area of inscribed circle) / (Area of circumscribed circle)
= (πs^2/4) / (πs^2/2)
= (πs^2/4) * (2/πs^2)
= 1/2
Therefore, the ratio of the area of the inscribed circle to that of the circumscribed circle is 1 : 2, which corresponds to option D.
Let's denote the side length of the square as "s". Since the square is inscribed in a circle, the diagonal of the square will be equal to the diameter of the circumscribed circle. The diagonal of the square can be found using the Pythagorean theorem:
Diagonal of square = √(s^2 + s^2) = √2s
Area of the square = s^2
- Inscribed Circle:
The diameter of the inscribed circle is equal to the side length of the square. Therefore, the radius of the inscribed circle is half the side length of the square, or s/2.
Area of the inscribed circle = π(s/2)^2 = πs^2/4
- Circumscribed Circle:
The diameter of the circumscribed circle is equal to the diagonal of the square, which we found to be √2s. Therefore, the radius of the circumscribed circle is half the diagonal of the square, or (√2s)/2 = √2s/2.
Area of the circumscribed circle = π(√2s/2)^2 = π(2s/4) = πs^2/2
- Ratio of Areas:
Now we can find the ratio of the area of the inscribed circle to that of the circumscribed circle:
Ratio = (Area of inscribed circle) / (Area of circumscribed circle)
= (πs^2/4) / (πs^2/2)
= (πs^2/4) * (2/πs^2)
= 1/2
Therefore, the ratio of the area of the inscribed circle to that of the circumscribed circle is 1 : 2, which corresponds to option D.
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A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?a) 2 : 3b) 3 : 4c) 1 : 4d) 1 : 2e) None of theseCorrect answer is option 'D'. Can you explain this answer?
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