Area of a quadrilateral knowing the lengthes of its sides and diagonals

Question

I have the length of four sides and two diagonals.

Sides' lengthes are: AB 26ft; BC 36ft; DA 27.4; CD 35.8ft

Diagonals' lengthes are AC 37.8ft; BD 50.6

I have used this formula to find the area I got 950sq.ft as the answer.

When I try to check in this site I am getting different answer.

Please help me to find out the correct area. I don't know the angles

I would suggest dividing into two triangles using the diagonal $AC$ and then using Heron's formula for each triangle. — André Nicolas, Apr 14 '16 at 05:05
And then you can do the same thing using the diagonal $BD$ to check the consistency of your lengths. — joriki, Apr 14 '16 at 06:22
Taking, in Brettschneider formula, in this order $a = 26; b = 36; c = 35.8; d = 27.4; p = 37.8; q = 50.6$, I find $956.08$. — Jean Marie, Apr 14 '16 at 06:26
Or you can use Brahmagupta formula after finding the angle in between the diagonal. — N.S.JOHN, Apr 14 '16 at 06:27

score 0 · Answer 1 · answered Jul 11 '20 at 11:29

Let us apply Bretschneider's Formula with $a=26$, $b=36$, $c=35.8$, $d=27.4$, $p=37.8$, and $q=50.6$. This formula can be applied to "a general quadrilateral" [1]. I will remark on this in a moment as it relates to the website you used. \begin{align} K&=\frac{1}{4}\sqrt{4(37.8)^2(50.6)^2-((36)^2+(27.4)^2-(26)^2-(35.8)^2)^2}\\ &=\frac{1}{4}\sqrt{4(1428.84)(2560.36)-(1296+750.76-676-1281.64)^2}\\ &=\frac{1}{4}\sqrt{14633379.1296-(89.12)^2}\\ &=\frac{1}{4}\sqrt{14625436.7552}\\ &\approx956.080434482 \end{align} Thus, the area of the quadrilateral is approximately $956.08$ square feet. This is the same result that user Jean Marie found in the comments. Now, if you exclude the length of the diagonal from $A$ to $C$ and enter all the information into the website you shared, being sure to select "feet" at the top of the form, the result of the calculations will be "$860.17$ square feet." If you do it once more, but with the length of the other diagonal provided instead, the result of the calculations will be "$914.54$ square feet." The website says that "this tool assumes the shape is convex, not concave." It does not appear to allow you to enter more than one field of information in the bottom portion of the form. Therefore, with all of this considered, I would not trust this website with answering this problem. The formula we used is far more applicable. Now, as it pertains to user N.S.JOHN's comment, Brahmagupta's Formula "is a special case giving the area of a cyclic quadrilateral" [2]. We don't know if this quadrilateral can be inscribed in a circle, so we don't know if it can be safely applied here in this problem, whereas Bretschneider's Formula "works on any quadrilateral, whether it is cyclic or not" [3].

Area of a quadrilateral knowing the lengthes of its sides and diagonals

1 Answers1