For a triangle,$a = 15 \, cm$,$b = 20 \, cm$ and $s = 31 \, cm$,find the value of $c$.

In quadrilateral $PQRS$,$PQ = 27 \, cm$,$QR = 30 \, cm$,$RS = 52 \, cm$,$SP = 53 \, cm$ and $PR = 51 \, cm$,then find the area of the quadrilateral in $cm^2$.

$A$ rectangular plot is given for constructing a house, having a measurement of $40 \, m$ long and $15 \, m$ in the front. According to the laws, a minimum of $3 \, m$ wide space should be left in the front and back each, and $2 \, m$ wide space on each of the other sides. Find the largest area where the house can be constructed (in $m^2$).

The lengths of the sides of a triangle are $13 \, cm$,$13 \, cm$,and $10 \, cm$. Find its area. Also,find the length of the altitude from the opposite vertex to the side whose length is $10 \, cm$.

The perimeter of a triangular field is $420\, m$ and its sides are in the ratio $6: 7: 8$. Find the area of the triangular field (in $m^2$).

Find the area of a triangle whose sides are $13\, cm$,$14\, cm$,and $15\, cm$. (in $, cm^2$)