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The area of a square is ‘d’. What is the area of the circle which has the diagonal of the square as its diameter?
- a)πd
- b)πd2
- c)(1/4) πd2
- d)(1/2)πd
Correct answer is option 'D'. Can you explain this answer?
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The area of a square is ‘d’. What is the area of the circl...
One important observation to solve the question :
Diagonal of Square = Diameter of Circle.
Let side of square be x.
From Pythogorous theorem. Diagonal = √(2*x*x)
We know area of square = x * x = d
Diameter = Diagonal = √(2*d)
Radius = √(d/2)
Area of Circle = π * √(d/2) * √(d/2) = 1/2 * π * d
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The area of a square is ‘d’. What is the area of the circl...
Explanation:
The diagonal of a square divides the square into two congruent right triangles. The length of the diagonal can be calculated using the Pythagorean theorem.
Calculating the diagonal of the square:
Let the side of the square be 's'.
Using Pythagorean theorem,
\(d^2 = s^2 + s^2 = 2s^2\)
\(d = \sqrt{2}s\)
Calculating the radius of the circle:
The diameter of the circle is the diagonal of the square,
\(d = \sqrt{2}s\)
Radius of the circle, \(r = \frac{d}{2} = \frac{\sqrt{2}s}{2}\)
Calculating the area of the circle:
Area of the circle, \(A = \pi r^2\)
Substitute the value of 'r',
\(A = \pi (\frac{\sqrt{2}s}{2})^2\)
\(A = \frac{\pi}{2} s^2\)
Comparing the areas:
Given that the area of the square is 'd',
Area of the square, \(d = s^2\)
\(s = \sqrt{d}\)
Substitute 's' in the area of the circle formula,
\(A = \frac{\pi}{2} (\sqrt{d})^2\)
\(A = \frac{\pi}{2} d\)
Therefore, the area of the circle which has the diagonal of the square as its diameter is \((1/2) \pi d\). Option 'D' is the correct answer.
The diagonal of a square divides the square into two congruent right triangles. The length of the diagonal can be calculated using the Pythagorean theorem.
Calculating the diagonal of the square:
Let the side of the square be 's'.
Using Pythagorean theorem,
\(d^2 = s^2 + s^2 = 2s^2\)
\(d = \sqrt{2}s\)
Calculating the radius of the circle:
The diameter of the circle is the diagonal of the square,
\(d = \sqrt{2}s\)
Radius of the circle, \(r = \frac{d}{2} = \frac{\sqrt{2}s}{2}\)
Calculating the area of the circle:
Area of the circle, \(A = \pi r^2\)
Substitute the value of 'r',
\(A = \pi (\frac{\sqrt{2}s}{2})^2\)
\(A = \frac{\pi}{2} s^2\)
Comparing the areas:
Given that the area of the square is 'd',
Area of the square, \(d = s^2\)
\(s = \sqrt{d}\)
Substitute 's' in the area of the circle formula,
\(A = \frac{\pi}{2} (\sqrt{d})^2\)
\(A = \frac{\pi}{2} d\)
Therefore, the area of the circle which has the diagonal of the square as its diameter is \((1/2) \pi d\). Option 'D' is the correct answer.
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The area of a square is ‘d’. What is the area of the circle which has the diagonal of the square as its diameter?a)πdb)πd2c)(1/4) πd2d)(1/2)πdCorrect answer is option 'D'. Can you explain this answer?
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