[Math] How to find the perimeter of a concave pentagon

algebra-precalculusgeometrypolygons

We know that a regular pentagon has five sides with identical lengths. But, the irregular pentagon has five sides with different angles. and moreover, the perimeter of a regular pentagon is 5a, where a is the side length of the regular pentagon. But how can I find out the formula of perimeter of an irregular concave pentagon? There is given a figure of a concave pentagon.

Figure of a Concave Pentagon

Here in the picture, the length of the AB and CD are Equal and the length of AB and BC are given. I have to find out the perimeter of the concave pentagon. I couldn't able to solve this anyway.

Best Answer

Hint: Notice how the figure looks like a rectangle with an equilateral triangle cut out.

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[Math] How to determine the side lengths of an irregular polygon when all interior angles are known

For a $n$ sided polygon, you need all the angles in order and $n-2$ consecutive side lengths in order to construct the polygon. So, you need the lengths of sides $B,C$ or $E,B$ or $D,E$ to construct your polygon.

The best way to find out the length of the remaining side is by drawing diagonals and applying triangle laws (sine or cosine rule).

Consider the (very badly drawn) pentagon. It is not drawn to scale, but you get the idea. Here are the steps you will take to find out the lengths of $D,E$.

1.Find out length of $X$ using cosine rule in $\Delta ABX$.

2.Knowing $\angle a,X,A,B$, find out $\angle e,\angle b$ using sine rule.

3.$\angle c = \angle\mbox{(between B,C)} -\angle b$. So, $\angle c$ is known.

4.Repeat the whole procedure for $\Delta CXY$. Find out $Y,\angle d, \angle f$.

5.$\angle g, \angle h$ are easily calculated now.

6.$\angle i$ is known. Apply sine rule in $\Delta DEY$ to find out $D,E$-the two unknown sides.

A (very crude) pentagon

[Math] How to calculate perimeter of Polygon with missing the length of one side

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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.

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Related Solutions

[Math] How to determine the side lengths of an irregular polygon when all interior angles are known

[Math] How to calculate perimeter of Polygon with missing the length of one side

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