The sides of a triangle are in the ratio of $\frac{1}{2}: \frac{1}{3}: \frac{1}{4}$. If the perimeter is $52 \text{ cm}$,then the length of the smallest side is ........ $\text{cm}$.

In the given figure, $ABCDEF$ is a regular hexagon whose side is $6\, cm$. $APF$, $QAB$, $DCR$, and $DES$ are equilateral triangles. What is the area (in $cm^2$) of the shaded region (in $\sqrt{3}$)?

$A$ circular park has a path of uniform width around it. The difference between the outer and inner circumferences of the circular path is $132 \, m$. Its width is.....$m$.

$AB$ and $CD$ are two parallel chords of a circle lying on the opposite side of the centre and the distance between them is $17 \, cm$. The lengths of $AB$ and $CD$ are $10 \, cm$ and $24 \, cm$ respectively. The radius (in $cm$) of the circle is

The ratio of the area of a sector of a circle to the area of the circle is $1:4$. If the area of the circle is $154 \, cm^2$,the perimeter of the sector is ........ $cm$.

The minute-hand of a clock is $14 \, cm$ long. The area covered by the minute hand in $30 \, min$ is.............$cm^2$