# How to calculate perimeter of Polygon with missing the length of one side?

I have following sides(PQRST) of a Polygon where PQ=13, QR=22, RS=8, ST=?, PT= 10 ... i need to find out ST? i don't have any angle i just have the shape? And for calculating perimeter i need to find out the ST length of polygon!

• Presumably, all the angles that look like right angles are supposed to be taken to be right angles, in which case a little insight and a use of Pythagoras should answer your questions. Mar 12, 2014 at 6:22
• I found its solution done by somebody using Pythagoras Theorem. and he got the missing side with length 13 and calculated perimeter 66 if am not wrong. But, i don't know how he has done. Because i don't have much knowledge about polygons and new here. I need some sort of a helping hand to solve it Mar 12, 2014 at 6:44

We can make use of the symmetry here and use Pythogoras theorm to Solve:

Observe that:

QR = PO = 22 {Opposite sides of a rectangle}
PT + TO = 22
TO = 22- PT = 22 - 10 = 12


Similarly,

PQ = OR = 13  {Opposite sides of a rectangle}
OS + SR = 13
OS = 13 - SR = 13 - 8 = 5


Now In right Traingle TOS, rt angled at O,

$TO^2 + OS^2 = ST^2 {Using Pythogoras Theorm}$
$12^2 + 5^2 = TS^2$
$TS^2 = 144 + 25 = 169$


Hence, TS = 13

Now you can find the perimeter.

• Great way to answer this :) Mar 12, 2014 at 6:57
• @Nomi thanks, if its useful, you can vote this up. Mar 12, 2014 at 6:58
• i did but,.. i need 15 reputation for doing it :P sorry Mar 12, 2014 at 7:12

Why don't you draw perpendicular from $T$ to $QR$ and from $S$ to $PQ$ ? Pythagoras Theorem.

Complete rectangles by drawing sides I told you about. Use the fact that opposite sides of rectangles are equal You should be able to get a right angled triangle with hypotenuse as $ST$ and other 2 sides as $13-8$ and $22-10$.

• I found its solution done by somebody using Pythagoras Theorem. and he got the missing side with length 13 and calculated perimeter 66 if am not wrong. But, i don't know how he has done. Because i don't have much knowledge about polygons and new here Mar 12, 2014 at 6:39
• @Nomi Get it now? Mar 12, 2014 at 6:45
• @Nomi Can't u just think of this polygon as some rectangles and a triangle put together and deleted all lines except the perimeter? You just gotta reconstruct those lines which question maker deleted. Mar 12, 2014 at 6:49
• thanks dear it solved the purpose :) Mar 12, 2014 at 6:54