Question

how do you use equations to solve probelems about the area of rectangles parallelegrams and trapozoids and triangles

Answer 1

Area of parallelograms

We can figure out a formula for the area of a parallelogram by dissecting the parallelogram and rearranging the parts to make a rectangle. Because the parallelogram and rectangle are composed of the same parts, they necessarily have the same area. (See the definition of area for more about why those areas are the same.)

We can see that they also have exactly the same base length (blue) and exactly the same height (green). Because base x height gives the area of the rectangle, we can use the same measurements on the parallelogram to compute its area: base x height. (As before, "height" is measured perpendicular to the base, and "base" is whichever side you chose first. See parallelogram.)

Area of triangle

Knowing how to find the area of a parallelogram helps us find the area of a triangle.

Dissecting the triangle

We can dissect the triangle into two parts -- one of them a triangle, and one of them a trapezoid -- by slicing it parallel to the base. If we cut the height exactly in half with that slice, the two parts fit together to make a parallelogram with the same base but half the height.

So base x half-height gives the area of the triangle. A similar dissection shows half-base x height. Either of them reduces to 1/2 bh.

The parallelogram's area is base x height, but that is twice the area of the triangle, so the triangle's area is 1/2 of base x height, as we saw with the dissection method.

(As always, pick a "base" and measure the height perpendicular to that base, from the base to the opposite vertex.)

hope this helped!