The pendant below symbolises the Deathly Hallows of the famous novel Harry Potter and the Deathly Hallows. A triangle inscribes a circle in the imitation below and its radius is $4 \mathrm{cm}$. The base segment of the triangular frame is divided into lengths of $6 \mathrm{cm}$ and $8 \mathrm{cm}$.

Find the perimeter of the pendant.

Two friends, as part of fun trip starts from a common point in two directions. Person $A$ went along $30° \mathrm{SE}$ with respect to the East and person $B$ went along $30° \mathrm{NE}$ with respect to the East.

Person $A$ travelled along a straight path till the periphery of a circular path, turned left, continued along the straight path to reach the centre of circular path, and paused. Person $B$ also reached the periphery of the circular path and asked person to join him.

Determine the angle person $A$ should turn at $C$, from his current direction to reach person $B$.

The figure below shows sketch of a base made around a large Gulmohar tree with cylindrical stem. The measure of its edges are given in the figure. 

The circumeference of the tree is $?$

In the picture below, the circumference of the front wheel is $2.2 \mathrm{m}$ and the distance from the centre of front wheel to pedal, $PO=70 \mathrm{cm}$. The distance from joint $B \mathrm{to} A$ on the handle bar is $5 \mathrm{cm}.$

Estimate the length of the rod $\mathrm{PA}.$

Below is the rough sketch of a roller with a handle to push over the soil and a stand to keep the roller steady when resting on the ground. Two rods are welded on the circular roller shield and their other ends are fixed at the centre of the handle. Two radial spokes each $50 \mathrm{cm}$ long, connect the roller shield with the centre of the roller. There is also a third rod $1.3 \mathrm{m}$ long, that connects the centre of the roller and the centre of the handle. 

The region shaded in the picture is to be covered by an iron sheet on both sides of the frame. Estimate the cost of an iron sheet if it costs $₹800 \mathrm{per} \mathrm{sq} \mathrm{m}$.

The figure below is a sketch of a circular auditorium surrounded by circular park. The centre of auditorium and the park is the same. The diameters of the auditorium and the circular path are $40 \mathrm{m}$ and $120 \mathrm{m}$ respectively. A person going for a walk from location $\mathrm{A}$ to $\mathrm{B}$.

Find the minimum distance the person has to walk.

In a fun experiment, two objects being made to move along circular path of radius $15 \mathrm{cm}$ at a uniform speed of $5 \mathrm{m}/\mathrm{s}$. The objects where released simultaneously and their locations on the path were $\mathrm{A} \mathrm{and} \mathrm{B}$. The objects collided at location $\mathrm{O}$ in $4$ seconds. Find the distance of the point of collision from the centre of the circular path.

The figure below shows a point sized wheel and a larger wheel. Both wheels are connected using chain, $OADBO$. The distance between the centres of the wheel is $25 \mathrm{cm}$ and radius of the larger wheel is $15 \mathrm{cm}$. Find the length of the belt.

Below is the design of a football trophy designed by a football club for their tournaments. A cavity is made at the base of a solid circular cone to fit a sphere of radius $13 \mathrm{cm}$ on its top. The diameter of the base of cone is $24 \mathrm{cm}$ and its height is $20 \mathrm{cm}$. Find the height of the trophy.