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, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.
What is the area (in cm²) of the rectangle PLMN?
Note: The figure shown is representative.
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius $r$ cm as shown in the figure. The side of the dodecagon is $d$ cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side $r$ cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.
- Which one of the following options has the correct sequence of objects arranged in the increasing number of mirror lines (lines of symmetry)?
- A circle with center at $(x, y) = (0.5, 0)$ and radius $= 0.5$ intersects with another circle with center at $(x, y) = (1, 1)$ and radius $= 1$ at two points. One of the points of intersection $(x, y)$ is:
- During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be $89.85^{\circ}$, the ratio of the Earth-Sun and Earth-Moon distances is closest to
