Other quadrilaterals

​ ​Maria is painting the stable where she spends her weekends. She finished the door and part of the outer wall. The top part of the gable wall and the sideboard of the stairs remain. Maria, of course ,wants to know what the surface is so that she can get enough paint. Both the top of the gable and the sideboard are types of slanted rectangles.

Maria starts with the sideboard of the steps. What kind of shape is that? If we turn it a little like this, you might recognize it. This is a parallelogram. The opposite sides are both parallel and equal.

Even though a parallelogram looks slanted compared to a rectangle, you actually calculate the area in exactly the same way, the base times the height. But how can that be right? It's completely slanted, so how can it be the same? It's like this, look at what happens when we cut the figure here. We get a triangle.

And if we move it over here, it fits properly and now it's a rectangle. The base is the same and the height is the same. So the base times height gives the area of both the rectangles and parallelograms. So when Maria measures the sideboard she must be careful about what is the base. It's 3 meters.

And then she measures the height, which makes the right angle with the base, and she gets 90 centimeters. The base times the height. Three times 0,9 meters equals 2,7 square meters. That's the siding. But what about the gable?

Is it a parallelogram as well? No. This is a different quadrangle. It has two opposite sides that are parallel and two that are not. The two opposite sides that are not parallel are of equal lengths.

This is an isosceles trapezoid. The area of a trapezoid is always calculated in the same way, whether it's isosceles or not. ​ ​There are two simple ways to do this. The first one is easy to understand but take some time to calculate. Do this, and you get a rectangle and two triangles. You can now calculate the area of each figure separately and add them together to get the isosceles, trapezoid's area.

But there's a faster way to calculate the area. First, look what happens when you take the base times the height, you get this area which is way too big, but there's actually another base, one that is shorter. A trapezoid has two bases, one short and one long. If you now calculate the area of the rectangle with the shorter base, you get this area, and this one is way too small. The fact is that you get the correct area of a trapezoid if you find a base that is exactly in the middle between the shorter and longer bases, and multiply it by the height.

Right here, you have a rectangle with the same area as a trapezoid. To calculate how long this base is, you do this, add the lengths of both bases and divide their sum by two, this gives you the average between the short and long bases, multiply this by the height. Maria has measured the gable of the stable and found that the lower, longer base is 5 meters, the upper shorter base is 3 meters, and the height is 2 meters. She puts the measurements into the formula and calculates the area to be 8 square meters, you get a parallelogram's area the same ways as rectangles, the base times the height. A trapezoid's area is calculated like this, the long base plus the short base, divided by two, times the height.