# Rectangle

## Meaning

### Informal

A flat shape with exactly four straight sides and four right angles. This includes squares (all sides the same) as well as oblong (non-square) figures that have only right angle corners.

These are all rectangles:

These are all not rectangles:

### Formal definition

On a plane, these statements are equivalent. That is, either could be taken as a definition, and the other would be a provable consequence of that definition. On a plane, the sum of the interior angles of a quadrilateral is 360°, so all the angles in an equiangular quadrilateral must be right angles, making the quadrilateral a rectangle. Going in the other direction, a parallelogram with any right angles will necessarily have all right angles, and so it is an equiangular quadrilateral.

But on a sphere, these definitions are not equivalent! On a sphere, the paths that we regard as "straight lines" are the great circles, and any two great circles must intersect, therefore no lines are parallel. Without parallel lines, parallelograms cannot exist and so, by the first definition, rectangles cannot exist. But equiangular quadrilaterals (and other polygons) are easy to create.

Incidentally, on a sphere, the four congruent angles will not be right angles. (But that's not part of the second "definition" and so it doesn't matter!) On a sphere, we can construct quadrilaterals with as many as three right angles, but cannot construct any with four (though the smaller the equiangular quadrilateral is, relative to the size of the sphere, the closer its angles will be to right angles).

## Emerging conceptions, misconceptions

When we ask a person to circle all the rectangles in some illustration, and they do not circle the squares, their image of rectangle is too narrow. It is important to know that squares are rectangles: they are special rectangles, in that their sides are all the same length.

If we ask people to draw a rectangle, and they draw a square, we can't be sure what they understand. They are correct, but have drawn a special case. (See examples.)

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## Etymology: What's in a word?

Latin Rect- + Angle

### Rectangles have Right Angles

A rectangle is a four-sided shape with only right angles. The word right has many meanings. One of its meanings is 'correct,' the opposite of 'wrong.' (When we get something wrong, we try to rectify our error.) Another meaning is the opposite of 'left.' Still another meaning is 'straight' or 'directly,' as in "After school, please come right home."

### Why all those meanings? What do they all have in common?

Like most words, right has great-…-great grandparents. The earliest ancestor we know might have sounded something like reg, and many of right's cousins are spelled like reg or rect. The original meaning was something like 'to move in a straight line.' When we regulate something, we are keeping it on track, keeping it from changing, keeping control. Rules and regulations keep people 'following the straight path,' doing the right thing. The word ruler is a cousin of regular and so it is a cousin of right. When the g in a word like regular is silent (as it is in right, and many other words), we get ruler. A ruler helps you draw a straight line. A king is a ruler, who makes rules that regulate the country. Tyrannosaurus Rex was 'king' of the dinosaurs; 'rex' is the Latin word for king, and is related to reg and rect. The king—royal is just regal with another silent g—was the richest, most powerful person, who gave people their rights, or took them away! When we name children Richard or Regina or, of course, Rex—or Henry or Roy with yet another silent g—we are comparing them to kings and queens!

### If right means something like 'straight,' why do we say right turn?

The word right suggests strength—not just the king but the right hand—and so a turn to the right is a turn toward that right hand.

### Where does a right angle get its name? Are there "left angles," too?

No, the right angle has nothing to do with right or left turns.

In this map, we say that Walnut Street makes a right angle with Main Street because it goes as 'straight' away as possible from Main Street. Corn Street veers off at a different angle.