Finding the Area of Any Quadrilateral

Quadrilaterals are four-sided, two-dimensional figures. There are numerous formulas that are taught for computing their areas, but they can all be found from a single source. Any quadrilateral can be divided into two triangles by drawing a line segment that connects two opposite corners (vertices). The area of a triangle should be a familiar one: one half of the base times the height. (The base is any one side of the triangle, and the height is the perpendicular distance from the base to the apex - the point of the triangle that does not lie on the base.) Since every quadrilateral can be depicted as the sum of two triangles, we have our generic formula for the area of a quadrilateral.

A = (1/2 * b1 * h1) + (1/2 * b2 * h2)

(b1 and h1 belong to the first triangle, b2 and h2 belong to the second triangle.)

This equation simplifies to easier to solve forms when we deal with special types of quadrilaterals. In particular, the Trapezoid (as it is known in the U.S.A. - it would be the Trapezium in Great Britain and elsewhere), parallelogram, rectangle and square are of interest.

Area of a Trapezoid = 1/2 * (b1 + b2) * h
Proof:
The trapezoid has two sides parallel to one another. If we choose the two parallel sides as the bases for our triangles, then each triangle has the same height - the distance between the parallel sides. Since h1 = h2, we can factor out the 1/2 and h from each triangle, leaving only the two bases to be added together.

Area of a Parallelogram (or Rhombus) = b * h
Proof:
The parallelogram has two pair of parallel sides. Opposite sides are also equal in length. When the parallelogram is divided into two triangles, the resulting triangles are congruent (equal in all ways - including area). Congruence is easily shown using either the SSS, SAS, or AAS criteria for triangles. Since both triangles have the same area, we simply double the area of one triangle to find the area of the whole parallelogram, giving us 2 * (1/2 * b * h). The rhombus is a special case of the parallelogram where all four sides are equal. This doesn't change the formula in any way.

Area of a Rectangle = L * W
Proof:
A rectangle is a special case of the parallelogram where all four angles are right angles. Since it is a parallelogram, A = b * h still. The fun part is that whichever side of the rectangle you choose to use as the base, the adjacent side is the height. Referring to the length (L) of one side of the rectangle and to the other as width (W), our b * h becomes L * W. (It doesn't matter which is which, though it is customary to call the longer side "length".)

Area of a Square = S^2
Proof:
A square is a special case of a rectangle - all its sides are equal in length. As a result, both length and width are equal to the length of one side (S). This gives us S * S for our area, also known as S^2.

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