# Calculate pentagon area based on lengths of all its sides

Sorry for this question. I guessed there is an online calculator to calculate the area of the pentagon if we know lengths of all its five sides. Actually there isn't

So, here are the lengths of sides of pentagon ABCDE: AB=19.14; BC=111.73; CD=33.23; DE=14.66; EA=110.75;

Q1. What is the area of this pentagon?

Also, there are more questions:

Q2. what's the formula to calculate the area of the pentagon based on its side lengths (side order counts)?

Q3. How many pentagons are possible to build if we know the side lengths? (of course, side order counts!)

Q4. Is this statement truthful: You can build the pentagon based on its side lengths then and only then when the side length is smaller then the sum of other sides, and it is also valid for all (other) sides.

Thank you.

• Are all inner angles equal to $120^\circ$? Otherwise, 'the' pentagon doesn't exists, since it isn't uniquely defined by the side lengths. Even if we may assume equal inner angles, there are some restrictions on the side lengths. – Ragnar Jul 21 '15 at 13:44
• This pentagon is not uniquely determined - there is more than one pentagon with these side lengths, and I bet they can have different areas. – Wojowu Jul 21 '15 at 13:46
• @Ragnar: There is no planar pentagon with "inner angles" equal to $120^\circ$. – Christian Blatter Jul 21 '15 at 15:04
• Oops, my bad. Should have been $180-72=108$ of couse – Ragnar Jul 21 '15 at 15:06

## 3 Answers

1. You can't tell. You can change the angles, and then the area will change.

2. From answer 1, there is no formula.

3. An infinite number (uncountable).

4. No.

Remember each polygon with $n$ no. of sides can be divided into $(n-2)$ triangles with a common vertex.

Thus for a pentagon $ABCDE$, all the sides are known but none of the angles is known hence a number of pentagons can be constructed using the side lengths given here & taking arbitrary values of interior angles. Hence, we can say

1. Area of pentagon $ABCDE$ can't be calculated in this case as it's uncertain
2. There is no formula to calculate area as pentagon is not unique
3. There are infinite no. of pentagons having sides as given & arbitrary interior angles
4. The statement is not true. No such unique pentagon can be constructed.

It would be possible to construct the pentagon $ABCDE$ & calculate its area if any two interior angles are given as an additional condition.

Well I suppose you could use www.wolframalpha.com which is always a good site.

I forgot the exact formula, but once I remember it I will edit it into this post.