Circles are drawn with four vertices as the centre and radius equal to the side of square. If the square is formed by joining the mid-points of another square of side $2\sqrt{6}$, find the area common to all the four circles.

ABCD is a circle and circles are drawn with AO,CO, DO and OB as diameters.Areas E and F are shaded. E/F is equal to

The diagram shows six equal circles inscribed in equilateral triangle $ABC$. The circles touch externally among themselves and also touch the sides of the triangle. If the radius of each circle is $R$, area of the triangle is

A boy Mithilesh was playing with a square cardboard of side $2$ meters. While playing, he accidentally sliced off the corners of the cardboard in such a manner that a figure having all its sides equal was generated. The area of this eight-sided figure is:

Let $P_1$ be the circle of radius $r$. $A$ square $Q_1$ is inscribed in $P_1$ such that all the vertices of the square $Q_1$ lie on the circumference of $P_1$. Another square $Q_2$ is inscribed in the circle $P_2$. Circle $P_3$ is inscribed in the square $Q_2$ and so on. If $S_N$ is the area between $Q_N$ and $P_{N-1}$ where $N$ represents the set of natural numbers. If the ratio of sum of all such $S_N$ to that of the area of the square $Q_1$ is $\frac{a-\mathrm{\pi }}{b}$ then $a+b=$?

In the figure, $ABCDEF$ is a regular hexagon and $PQR$ is an equilateral triangle of side '$a$'. The area of the shaded portion is $23\sqrt{3} c{m}^2$ and $CD:PQ::2:1$.

If the area of the circle circumscribing the hexagon is $X\pi c{m}^2$ then $X=?$

Let $S_1, S_2, ...$ be the squares such that for each $n\ge 1,$ the length of the diagonal of $S_n$ is equal to the length of the side of $S_{n+1}$. If the length of the side of $S_3$ is $4 cm$. What is the area of the square $S_{11}$?