Formula for the Area of a General Quadrilateral

[196]

I'm looking around the Internet for formulas for the area of a general quadrilateral (preferably with a proof or at least a diagram), and I find this sort of thing:
Finding the area of quadrilaterals . I'm both horrified and curious about the author's motiviation.

Then there is this one:
Proof of area of quadrilateral
At first it seems OK (if overly complicated), but then procedes to "jump the shark" with the introduction of "K" between (12) and (13). Am I missing something? This really isn't a "proof". It is about half a proof, then it introduces the very formula that it is attempting to prove into the mix.

I'm trying to work out a proof for the area of a general quadrilateral based on triangles, using at most Herron's Forumula, the Cosine Law, and some trig identities. This would make it a suitable project for advanced high school students.

I'm interested in your comments, insights, and ideas

 

[Vorpal]

Historical note: this is a generalization of Brahmagupta's formula (if the quadrilateral is inscribed in a circle, the cosine term vanishes), which was probably known by Heron as well. If p,q are the diagonals, the term involving cosine can also be replaced by (1/4)(ac+bd+pq)(ac+bd-pq).

I'm not sure what you think the problem with the proof is. If it's the introduction of K right before equation (13), then that part is completely unproblematic, being based on a decomposition of the quadrilateral into two triangles, the area of a triangle being half the product of two sides times the sine of the included angle.

 

[Gagglegnash]

Hi

For me, it's the Monte Carlo Method, all the way... but just because I like using my shotgun as a random-point generator.

 

[196]

I'm not sure what you think the problem with the proof is. If it's the introduction of K right before equation (13), then that part is completely unproblematic

The problem was me missing something obvious. I was blinkered by my expectation that the solution would involve Herron's Formula. Thanks for the push in the right direction.

My "jumping the shark" comment was wrong and completely unwarranted.

Thanks Vorpal!

 

[196]

Hi

For me, it's the Monte Carlo Method, all the way... but just because I like using my shotgun as a random-point generator.

So you suggest we could set up a target with a quadrilateral on it and my grade 11 students could take turns shooting it with a shotgun? Cool! We have firearms training at my school, so this is totally possible and I might even get to keep my teaching license.

 

[Skeptic Guy]

I'm looking around the Internet for formulas for the area of a general quadrilateral (preferably with a proof or at least a diagram), and I find this sort of thing:
Finding the area of quadrilaterals . I'm both horrified and curious about the author's motiviation.

Then there is this one:
Proof of area of quadrilateral
At first it seems OK (if overly complicated), but then procedes to "jump the shark" with the introduction of "K" between (12) and (13). Am I missing something? This really isn't a "proof". It is about half a proof, then it introduces the very formula that it is attempting to prove into the mix.

I'm trying to work out a proof for the area of a general quadrilateral based on triangles, using at most Herron's Forumula, the Cosine Law, and some trig identities. This would make it a suitable project for advanced high school students.

I'm interested in your comments, insights, and ideas

The first proof you refer to comes from a website called "Helium". Maybe a prerequisite is that the author has to be inhaling the referenced gas at the time?

 

[Vorpal]

No problem.

But uh... I've just tried to read your first link. I had to stop before my mind 'asploded. Either it was written by someone on LSD or someone has written an essay generator of this sort to be vaguely scientific and mathematical. All in all, I suppose other bits of wisdom from the same author are actually somewhat appealing the same sense as a Jabberwocky that doesn't rhyme.

 

[Gagglegnash]

Hi

...or you could simulate my shotgun with your computer's random-number generator, used to produce X-Y coordinates for the points on the sheet of paper (known area) on which the quadrilateral (unknown area) is drawn.

Another application of the Monte Carlo method is the one using an infinite number of points: Cut the quadrilateral out of the paper (hobby paper or cardboard, in this case). Measure the whole sheet's surface area (it's a rectangle, so it's easy... unless you were sloppy with the shears, in which case it's a wrecketangle, but still easy), weigh the whole sheet, then weight the cut-out quadrilateral. After that, it's a straight forward relationship of total weight/total area to cut-out weight/unknown area.

unknownArea = (quadrilateral.weight / wholeSheet.weight) * wholeSheet.area;

Thing is: You need fairly heavy paper and an accurate scale.

Challenge the class to figure out a way to get the ratio of whole-sheet to quadrilateral masses/weights. One idea might be taping a whole sheet of paper and the quadrilateral to either end of a meter stick and using the stick as a balance beam....

On the other hand, a well supervised demonstration of the shotgunnular Monte Carlo method, demonstrating the problems with clustering (standing too close so you blow a solid hole in the middle of the paper, making the result inaccurate and difficult to derive), insufficient data sampling (standing too far away and getting an insufficient number of data points to be statistically significant) and biasing (not, "aiming," well enough so the data points are insufficiently random) might be educational...

AND it would probably get the class's undivided attention.

 

[196]

Another application of the Monte Carlo method is the one using an infinite number of points: Cut the quadrilateral out of the paper...

I still like the shotgun idea, but this is probably a bit safer. Seriously, it's a great idea, thanks.

 

[CaveDave]

I'm looking around the Internet for formulas for the area of a general quadrilateral (preferably with a proof or at least a diagram), and I find this sort of thing:
Finding the area of quadrilaterals . I'm both horrified and curious about the author's motiviation.

[Snip]

I'm interested in your comments, insights, and ideas

[My bolding]

This is obviously the work of Professor Irwin Corey*,The Worlds Foremost Authority.
This article on Hammond Organ tonewheel generators, while putatively written under his monicker, may be a clever fake.

[excerpt]

"The tone generator has a base-plate of prefabulated amulite, surmounted by a malleable logarithmic casing in such a way that the two spurving bearings are in a direct line with the pentametric fan. The latter consists simply of six hydrocoptic marzelvanes, soffitted to the ambifacient lunar waneshaft in such a way that side fumbling is effectively prevented. The main windings are of the normal lotus/delta type placed in panendermic semi-boloid slots in the stators, every seventh conductor being connected by a non-reversible tremulant pipe to the differential girdle spring on the "up" end of the grammeters.

"Forty-one manestically spaced grouting brushes are arranged to feed HAMMOND Generator Oil into the rotor slipstream. This oil is a mixture of high 'S'-value phenylhydrabenzamine and 5% reminative 1,2,1-tetryliodihexamane. The oil has a high specific percosity when measured against P=2.5C^N^6.7 where 'N' equals the diathetical evolute of retrograde temperature/phase disposition and 'C' is Cholmondeley's annular grillage coefficient. Before adding oil to the generator, 'N' should be measured with the aid of a metapolar refractive pilfrometer, although adequate results can be obtained using a Hopper transcendental dadoscope.

*Often seen in the '60s on Johnny Carson and Ed Sullivan, among others. See him here.

Enjoy

Dave