12,3 - Types of Quadrilaterals

Links and useful resources

Concept summary and lesson

Quadrilateral > Trapezoid > Parallelogram > Rhombus > Square
Parallelogram > Rectangle > Square

Definitions

Quadrilateral: four-sided polygon
Trapezoid: Quadrilateral with one (or more) pair of parallel sides
Parallelogram: Quadrilateral with two pairs of parallel sides (parallel sides are also equal length, not adjacent obviously)
Rhombus: Parallelogram with all sides congruent (equal measure)
Square: Rhombus with all right angles
Rectangle: Parallelogram with all right angles
Kite: Quadrilateral with two pairs of equal-length sides, where the equal-length sides are adjacent.

About quadrilaterals

Quadrilateral means "four-sided" and it encompasses all polygons with four sides. We're going to be looking at convex quadrilaterals for now, which means that all of the interior angles are less than 180 (there are no "indented" corners that poke inside the shape).

We have different ways of grouping and classifying them:

How many sides are parallel?
- none: quadrilateral
- two: trapezoid
- four: rhombus
How many angles are right angles?
- 0-3: quadrilateral
- 4: rectangle
How many sides are equal in length?
- 4: rhombus
- 2 pairs (not equal to each other): kite or parallelogram. We don't talk much about kites... but they're quadrilaterals where there are two pairs of congruent sides, and the congruent sides are adjacent instead of opposite each other.

Think of the classification as a bunch of checkboxes: Any given quadrilateral can be several of these at once. For example, think of a square:

quadrilateral? check
parallelogram? check
rhombus? check
rectangle? check
kite? check

Areas of quadrilaterals

Any trapezoid has an area equal to $\frac{1}{2} h \cdot (b_{1} + b_{2})$ , where $h$ is the distance between the bases, and $b_{1}, b_{2}$ are the lengths of the bases.

Since rectangles are trapezoids, the same formula works there. But, since the sides are perpendicular, $h$ is just the length of one of the sides. The bases are also equal, so it ends up simplifying down to $w \cdot h$ (width * height) as you know from before!

A parallelogram is a trapezoid with equal bases, so it simplifies down to $b \cdot h$ , where $h$ is the perpendicular distance between two of the parallel sides, and $b$ is the length of one of those sides.

Shape	Area formula
Trapezoid	$\frac{1}{2} h \cdot (b_{1} + b_{2})$
Parallelogram	$h \cdot b$
Rectangle	$w \cdot h$

For other quadrilaterals, the best strategy is usually to cut them into pieces that you can compute easily and then add up the results.

Guided practice

Is every square also a rhombus? Is every rhombus also a square?
Explain how to arrange four 3-4-5 right triangles so that together they form a rhombus with side length 5.
$A B C D$ is a rhombus in which the diagonals have lengths $A C = 8$ and $B D = 12$ . In this problem, we’ll find the area of $A B C D$ .

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Homework

#hw (math) (gr7) 12.3 Types of Quadrilaterals (p. 457) 2025-03-26