Question

why do a rectangle and a parallelogram share the same formula?

why does the triangle have a formula of b*w /2

Answer 1

A rectangle is easy, as it can be divided up into a grid WxL and the area is just WL

Take a parallelogram ABCD with sides a and bases b. If you draw vertical altitudes at A and B, you will see a rectangle in the middle, a triangle sticking out on one end, and a triangle missing on the other end. The extra triangle exactly fits the space, and you end up with a rectangle, with the original base b, and height = h/a = sinA, with A the base angle.

Now, for a rectangle, A=90°, so area=base*height. For the parallelogram, it's base * side * sinA = base * height.

The same logic can be used to find the area of a trapezoid, which is just a rectangle with two triangles tacked on the ends. If the bases are a and b, area = (a+b)/2 * height.

A rectangle is just a trapezoid with bases a=b, so (a+b)/2 = b, and area = b*height.

A triangle is just a trapezoid with upper base a = 0, and (a+b)/2 = b/2, and area = h*b/2 = bh/2.

So, triangles, rectangles, and parallelograms are just special cases of a trapezoid.