Find Area of Trapezium with Given Sides and No Width

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In summary, the area of a trapezium can be found by breaking it up into two triangles and a rectangle, using the general rule for trapezoids, or using properties of right triangles formed by cutting out the center rectangle. There is no simple formula for finding the area, but it is possible to solve for the height by using Heron's formula or other methods. However, the trapezoid may not be unique and the area can vary depending on the angle of the corners.
  • #1
sniffer
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how can you find the area of trapezium for which all area are given but no width?

is there a simple formula?

see the attached image.
 

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  • #2
I don't know what width you are referring to. You could break it up into two triangles and a rectangle and sum up the areas, or use the general rule for trapezoids:

[tex] A = \frac{b_1+b_2}{2}h [/tex] where b's represent the parallel sides.
 
  • #3
I do not know of a simple formula, but you can solve for h by using properties of the right triangles formed by cutting out the center rectangle.
 
  • #4
Without knowing more than just the lengths of the sides (in order to use whozum and integral's ideas you would need to know at least some of the angles at which the sides join). Imagine this trapezoid made from sticks pinned together so that the sides can rotate on the pins (you can change the angles). You can flex the trapezoid to get a number of different area with the same side lengths.
 
  • #5
that's right.

i think so. the information given does not give a unique trapezium.
you can keep on shearing it until you get zero area.

that's why it looks odd.

thanks.
 
  • #6
no it is possible to do what he says.
Suppose that we "slide" the right side of the trapezium along the upper width until it meets the starting point of the other side. Now we have a triangle with sides(speaking for the numerical example)4,6,7. We can find this one's area by using

[tex]\sqrt u(u-a)(u-b)(u-c) [/tex]

where u is (a+b+c)/2 and a,b and c are the sides of the triangle. Now that we know the area of the triangle we can find its height. This is also the height of the trapezium so we have the area of the trapezium
 
  • #7
I have provided a jpeg file to make myself more clear on what I'm talking about
 

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  • #8
it actually is possible,but the formula is complicated
 
  • #9
it is definitely possible and pretty easy but there is no such formula or it maybe be complicated.
 
  • #10
http://home.comcast.net/~Integral50/Math/trapezoid.pdf [Broken] is the bulk of the alegbra.
 
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  • #11
As long as the parralel sides [itex]a[/itex] and [itex]c[/itex] are of different lengths, you can use Heron's formula to find the height, and from there it's easy.
If the parralel sides are of the same length, then the area is not determined by the side lenghts.
 
  • #12
stupidkid said:
it is definitely possible and pretty easy but there is no such formula or it maybe be complicated.

If that was an answer to my post, there exists such a formula and it is called Heron's Formula. It is used to find the area of a triangle only by using the sides. What I did is clear and correct. Integral provided just another way to find the area.

If you have a look at my attachment in my previous post you will understand what I mean.
 
  • #13
thanks a lot for the help.

actually my friend asked me this question, and i said to him
it is not possible because the trapezium is not unique.

heh he. i am wrooong...

thanks.
 
  • #14
sniffer said:
thanks a lot for the help.

actually my friend asked me this question, and i said to him
it is not possible because the trapezium is not unique.

The trapezoid is not necessarily unique. You are correct:

For example, consider a trapezod with side lenghts 1,1,1, and 1. It's quite easy to see that any quadrilateral with side lengths like that is going to be a trapezoid . However, since this trapezoid happens to be a parralelogram we know that the area of that trapezoid is going to be [itex]\sin\theta[/itex] where [itex]\theta[/itex] is the one of the corner angles.

Since the corner angle can vary, so can the sine of it.

There is also an IMHO pedantic argument that given just four numbers it's not necessarily possible to identify which pair of sides is parralel.
 

1. How do you find the area of a trapezium with given sides and no width?

The formula for finding the area of a trapezium with given sides and no width is A = 1/2(a + b)h, where a and b are the lengths of the parallel sides and h is the height of the trapezium.

2. Can you explain the formula for finding the area of a trapezium in more detail?

The formula for finding the area of a trapezium is derived from the formula for finding the area of a parallelogram, which is base x height. In a trapezium, the two parallel sides are considered the bases, and the height is the perpendicular distance between the two bases. Since the trapezium has two parallel sides, we take the average of those sides and multiply it by the height to get the area.

3. What if I don't know the height of the trapezium?

If you do not know the height of the trapezium, you can use the Pythagorean theorem to find it. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. You can use this theorem to find the height of the trapezium if you know the lengths of the two parallel sides and the width of the trapezium.

4. Is there a specific unit of measurement for the sides and height of a trapezium?

No, there is no specific unit of measurement for the sides and height of a trapezium. You can use any unit of measurement, such as inches, centimeters, or feet, as long as you use the same unit for all the measurements in the formula.

5. Can the formula for finding the area of a trapezium be used for any type of trapezium?

Yes, the formula for finding the area of a trapezium can be used for any type of trapezium, regardless of the shape of the bases or the length of the sides. As long as you know the lengths of the parallel sides and the height, you can use the formula to find the area of a trapezium.

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