$\begingroup$

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.

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.

$\endgroup$ 1

6 Answers

$\begingroup$

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

$\endgroup$ 4 $\begingroup$

Draw a line through $E$ parallel to $BC$. What can you say about the triangles thus formed?

$\endgroup$ $\begingroup$

The triangle $AED$ is equilateral and its base is $AD=BC=l$ so the perimeter is $3l+AB+CD$.

$\endgroup$ 3 $\begingroup$

We know the triangle $AED$ is an equilateral triangle. Therefore $AD=AE=DE=BC$. Thus the perimeter is:$$P=3BC+2AB$$

$\endgroup$ $\begingroup$

Perimeter of the retangle: 2AB + 2BC

Perimeter of the equilateral triangle: 3BC

Add the two together and remove the two BC lines (left edge of the retangle, left edge of the triangle): 2AB + 3BC

$\endgroup$ $\begingroup$

The perimeter formula is 5a. Since it's a square with an inscribed equilateral triangle, the other two (triangle) sides are equal to the 3rd -- and the remaining 3 sides of the square.

3 sides of a square + 2 of a triangle

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy