Quadrilaterals are just the four-sided shapes (“quad” meaning four); they include:
Rectangles: four sides connected by four 90-degree angles
Rectangle Area Formula:
Squares: four sides of equal length connected by four 90-degree angles.
Square Area Formula:
Parallelograms: four-sided figure where opposite sides are parallel
Parallelogram Area Formula:
Parallelograms also have the nice property that opposite angles are of the same degree, and opposite sides are the same length.
Rhombus: a parallelogram with equal sides
Rhombus Area Formula:
Trapezoids: four-sided figure with one pair of parallel opposing sides
Trapezoid Area Formula:
Now a lot of these definitions overlap with others, so here is a diagram that explains how all of these shapes relate to one another:
Finally, one fact that is sometimes helpful in solving problems with quadrilaterals is:
Quadrilaterals Have 360 Degrees: The sum of the angles in a quadrilateral is 360.
This fact will come up again when we talk about finding the values of unknown angles.
1. Find the area of the entire figure below.
Note that since and , we know that is a parallelogram. Thus, by the area formula above, we know that its area is Now, we also know by the Reverse Pythagorean Theorem that and since a straight line has 180 degrees, we conclude that Thus, we can apply the Pythagorean Theorem to get that Thus, the area of triangle equals and the total area of the figure is .
2. Find the area of the entire figure below.
Imagine drawing a line in the above diagram here:
Then, the left-hand shape must be a rectangle (since quadrilaterals must have angles that add up to 360 degrees and the other three right angles take up 270 degrees). Thus, as rectangles are a kind of parallelogram, the top side of the rectangle must be 7 units long. Thus, the top side of the right-hand figure must be 3 units long, and of course, that figure must also be a rectangle. Thus, we can find the area of the whole shape as:
3. The area of the below parallelogram is 30, is perpendicular to , and the ratio of the area of to the area of is 1 to 3. What is the value of ?
Since the area of is 30, we get that which implies Now, the area of will be whereas the area of will be . Thus, by the given ratio of those areas, we conclude that . And since we get that and so We can then use the Pythagorean Theorem to conclude
4. The dotted lines below meet at a right angle. Find
Imagine adding the following lines to the figure:
Then, we get that and thus that . So we can find by the Pythagorean Theorem, and it will equal 5.