Can You Count Them All?

Probability Level 2

How many rectangles are there?

16 10 22 25 28

1 solution

Eli Ross Staff
Oct 19, 2015

We can break these up into two types.

First Type: Rectangles in a single "column". Note that if a column has $n$ rectangles, then there are $n$ individual rectangles, then $n-1$ rectangles by combining 2 adjacent ones, and so on. There are 20 of these types:

Second Type: Rectangles that span multiple "columns". These are easy to just count; there are 5 of them.

Its not an overly clear picture. I choose 16, some of the shapes are/look very close to squares.

Kevin Bishop - 5 years, 7 months ago

In fact, squares are a special type of rectangle. A rectangle is a Quadrilateral with 4 right angles and equal-length opposite sides.

Eli Ross Staff - 5 years, 7 months ago

This is a very misleading question then. Common sense tells us that a square is not a rectangle. How are we supposed to know wh.ich perspectibe we need to look at this from

Ethan Czereda - 5 years, 7 months ago

@Ethan Czereda – A square is just a special type of a rectangle, as mentioned above.

You can think of this similarly to real numbers. For example, an integer (e.g., 2) is just a special case of a real number. If you were asked how many real numbers were in $\left\{\pi,\ 2,\ \frac{1}{2},\ \sqrt{-1}\right\}$ , you would definitely count the 2!

Eli Ross Staff - 5 years, 7 months ago

@Ethan Czereda – Definitions: A Rectangle is an equiangular Quadrilateral. A Square is an equiangular, equilateral Quadrilateral. Therefore: All square are Rectangles. Some Rectangles are Squares.

btw: It is also true that : All Squares are Rhombi. A Rhombus is an equilateral Quadrilateral.

Mathematics isn't based on common sense. If you know the definitions and logic you will do all right.

Robert Lucas - 5 years, 7 months ago

Different perspectives on this I guess Eli. I prefer to define all of rectangles, rhombi, squares, parallelograms, trapezoids, isosceles trapezoids and kites* as special quadrilaterals.
Some texts will DEFINE rectangles, rhombi and squares as special parallelograms - not my choice. Doesn't really include their quadrialateralness in the definition and I think it belongs there. *Definitions of kites as a special quadrilateral seem to vary. A Kite is a quadrilateral with two distinct pairs of congruent, consecutive sides. This allows squares to be kites (two distinct ways the count the pairs of sides). A Kite is a quadrilateral with two distinct pairs of congruent, consecutive sides
but is not equilateral. This definition does not allow squares to be a subset of kites. I prefer the first. Your definition of rectangle goes beyond the definition (equiangular) Four right angles and opposite sides congruent are properties derivable from that definition.

I am sorry - I have strayed way off the original problem. They are a favorite of mine. I prefer a chart of the special quadrilaterals to show their relationships to each other. When I do Venn diagrams I use colored borders or cross hatched interiors.

Robert Lucas - 5 years, 7 months ago

Can You Count Them All?

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