Formula for finding the area of quadrilaterals?

Question

I'm curious about the area of quadrilaterals when the measures of the sides are known. I know that the formula for a triangle is the square root of (half perimeter-s1)(half perimeter-s2)(half perimeter-s3). Is a quadrilateral harder?

Answer 1

The formula you give is called Heron's Theorem, which is technically a case of Brahmagupta's Theorem. Brahmagupta's Theorem is for cyclic quadrilaterals, meaning ones that can be inscribed in circles. The theorem is
Area = sqrt ( (s-a)(s-b)(s-c)(s-d) ) where s is the semiperimeter and a,b,c,d are the sides. When d=0, that gives you a triangle (actually there's an extra semiperimeter in your formula). Anyway, Brahmagupta's only works for cyclics.

For non-cyclic quadrilaterals, there is no one formula. For trapezoids the formula is A = 1/2*h*(b1 + b2) where b1 and b2 are the parallel bases, h is the distance between them. For a kite, A = d1*d2 where those are the diagonals. For any parallelogram, A = l*w (length and width). For more irregular shapes, the best thing to do is split them up into triangles and work it out that way.

Hope that helps you!

Answer 2

A square is a quadrilateral. All sides are the same length. S (side) squared is the area. For a rectangle, the area is defined by length times width. For other quadrilaterals, the formulas are somewhat similar, but for an irregular quadrilateral (the shape of an arrow point or something else that is an odd shape) can be divided into triangles in order to find the area.

Answer 3

If you want a method that will solve any quadrilateral, and not just parallelograms or rectangles or trapezoids, then yes, it's harder. The best way is to divide the quadrilateral into triangles, find the areas of the triangles, and add them together. This isn't always easy, since without known angles or something, it's difficult to find the altitudes of the triangles.

For certain types of quadrilaterals, the area is easier to find. For a paralellogram (any quadrilateral with opposite sides parallel to each other), the area is (length of base)*(height measured perpendicular to the base). The area of a trapezoid (two opposite sides parallel, the other two not) is 1/2*(height)*(sum of the lengths of the bases, or the parallel sides). Hope this helps!

Answer 4

there can not be a formula that works for any quadrilateral, since it could have a reflex angle(> than 180) at one of the vertices ( unlike a triangle which by definition can not), that vertex would make that corner concave in, in other words you can have two quadrilaterals with congruent sides but different angles. Visualization mode: get four straws of dif sizes connect them at the endpoint and then bend the angles you will see that the area varies and the measurment of the sides does not.

Answer 5

there are different formulas for different types of quadrilaterals (squares, parallelograms, trapezoids, rectangles, kites etc.)
its just a matter of memorizing them
they're all manly based on the formula basexheight=area
e.g. area of a rectangle= BxH
area of a parallelogram=BxH
area of a square = S^2 (or BxH since the sides are equal, its just shorthand to say it as side sq.
area of a trapezoid=1/2H(B1+B2)
this one is a little different as the trapezoid doesn't have opposite congruent parallel sides as a parallelogram does

area of a kite= (area of isosceles triangle)+ (area of second isosceles triangle)
kites are defined as a quad. w/ two pairs of congruent sides, that are not parallel and are made of isosceles triangles, so the easiest way is to identify the two triangles (share a common base) to find the area

so basically, its mostly memorization and knowing the properties of each quad.

Answer 6

I thought the formula for a triangle was 1/2*base*height? I don't know where the square root is coming from.