The diameters of two circles are the side of a square and the diagonal of the square. The ratio of the areas of the smaller circle and the larger circle is :
এই প্রশ্নটি সহ গণিত বিষয়ের আরও কিছু গুরুত্বপূর্ণ প্রশ্ন আছে
১ + ৫ + ৯ + ........... +৮১ = ?
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<b>Question:</b> The diameters of two circles are the side of a square and the diagonal of the square. The ratio of the areas of the smaller circle and the larger circle is:
Given that the correct answer is <b>$\frac{1}{2}$</b>, let's delve into the detailed reasoning behind this:
<h3>1. Understand the Problem</h3>We have two circles with their diameters equal to the side and the diagonal of a square, respectively. Let's define the side of the square as <code>s</code>.
<h3>2. Calculating the Diameters</h3>- The diameter of the smaller circle is the side of the square, which is: <br> $d_{small} = s$
- The diameter of the larger circle is the diagonal of the square. Using the Pythagorean theorem, the diagonal <code>d</code> of a square with side <code>s</code> is given by: <br> $d_{large} = \sqrt{s^2 + s^2} = s\sqrt{2}$
<h3>3. Calculating the Radii</h3>Next, we find the radii of the circles. The radius is half of the diameter: <br> Radius of the smaller circle: <br> $r_{small} = \frac{s}{2}$ <br> Radius of the larger circle: <br> $r_{large} = \frac{s\sqrt{2}}{2}$
<h3>4. Calculating the Areas</h3>The area of a circle is given by $A = \pi r^2$. Therefore: <br> Area of the smaller circle: <br> $A_{small} = \pi \left(\frac{s}{2}\right)^2 = \pi \frac{s^2}{4}$ <br> Area of the larger circle: <br> $A_{large} = \pi \left(\frac{s\sqrt{2}}{2}\right)^2 = \pi \frac{(s^2 \cdot 2)}{4} = \pi \frac{2s^2}{4} = \pi \frac{s^2}{2}$
<h3>5. Finding the Ratio of the Areas</h3>To find the ratio of the area of the smaller circle to the larger circle, we divide the area of the smaller circle by the area of the larger circle: <br> $\text{Ratio} = \frac{A_{small}}{A_{large}} = \frac{\pi \frac{s^2}{4}}{\pi \frac{s^2}{2}} = \frac{\frac{s^2}{4}}{\frac{s^2}{2}} = \frac{1}{2}$
<h3>Conclusion</h3>Hence, the ratio of the areas of the smaller circle to the larger circle is <b>$\frac{1}{2}$</b>.
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