Square Area and Diagonal Calculation
Let the side length of the square be $s$. The area of the square is given as $d$. Therefore, we have:
- $s^2 = d$
The diagonal of a square ($D_{sq}$) is related to its side length $s$ by the formula $D_{sq} = s\sqrt{2}$. Since $s = \sqrt{d}$, the diagonal is:
- $D_{sq} = \sqrt{d} \times \sqrt{2} = \sqrt{2d}$
Circle Properties from Square Diagonal
The problem states that the diagonal of the square serves as the diameter of the circle ($D_{circle}$).
- $D_{circle} = D_{sq} = \sqrt{2d}$
The radius ($r$) of the circle is half its diameter:
- $r = \frac{D_{circle}}{2} = \frac{\sqrt{2d}}{2}$
Calculating Circle Area
The area of a circle ($A_{circle}$) is calculated using the formula $A_{circle} = \pi r^2$. Substitute the value of $r$ we found:
- $A_{circle} = \pi \left( \frac{\sqrt{2d}}{2} \right)^2$
- $A_{circle} = \pi \left( \frac{2d}{4} \right)$
- $A_{circle} = \pi \left( \frac{d}{2} \right)$
- $A_{circle} = \frac{1}{2}\pi d$
Thus, the area of the circle is $\frac{1}{2}\pi d$.
Important Questions from Mensuration and Geometry
In the given figure, $P, Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45^\circ$. The figure is representative.
The area of the shaded region $PQRO$ is ______________ cm$^2$.
- A straight line $y = x - 1$ intersects a circle with center at $x = 1, y = 1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is _______. (rounded off to three decimal places)
- The shell of a hollow spherical nanoparticle has a uniform thickness of 3 nanometers (nm). The outer radius of the nanoparticle is 5 nm. The ratio of the volume of the shell to the volume of the hollow core is ________
(Round off to one decimal place) - The volume of a sphere of diameter 1 unit is ______ than the volume of a cube of side 1 unit.
- A window is made up of a square portion and an equilateral triangle portion above it. The base of the triangular portion coincides with the upper side of the square. If the perimeter of the window is 6 m, the area of the window in $m^2$ is ______
