Complements #3 : Properties of Shapes


This may seem like a huge tide turn from the rather complicated Sine, Cosine and Tangent posts previously, but it’s an area that is relatively simple, yet often taught incorrectly. The devil is in the detail.

#1 The Parallelogram

This seems to cause all sorts of errors, particularly spelling (check out the picture above!). I’m speculating it’s partially due to the ‘well it’s so straight forward I don’t need to check’ frame of mind. Perhaps.

Anyway, here are the most common errors I have encountered.

1. All four sides can’t be equal, it has two ‘sets’ of equal sides.

Noop. The second part is right, but that doesn’t imply that all four sides can’t be equal.

2. Squares are squares, not parallelograms.

Noop. Squares fall into all kinds of categories. They definitely ARE parallelograms.

3. Rectangles aren’t parallelograms, they’re rectangles.

Noop. Half truth. Rectangles are indeed rectangles, the rest is wrong.

4. Diamonds aren’t parallelograms.

Whoah whoah whoah whoah sweet child o’mine. Diamonds aren’t a maths thing, they’re a girl’s best friend. Us men, we get dogs. Unlucky. Anyway, stop that right now. If you mean this thing…


a… RHOMBUS. Then no, you’re wrong. It’s still a parallelogram. :p

So what makes a parallelogram a parallelogram? (*did I just use Epanalepsis there? I’m not sure)

Well, here’s a weird thing. Wikipedia (rightly) states that squares, rectangles and rhombi are parallelograms, yet look at their list of properties:


Pretty sure a square has rotational symmetry of order 4, not 2.

Anyhoo, Wolfram is a bit more reliable and says this instead:


That’s better methinks.

As a random aside, did you know that if you connect the midpoints of ANY quadrilateral, regular or irregular, you form a parallelogram? Weird.

parallel1 nonquadpar

#2 The Square

As I hinted at earlier, the square falls under many different classifications.

Technically speaking it is a…

square, parallelogram, rhombus, rectangle and a quadrilateral,

Maybe this will help:


There’s a curious debate as to whether parallelograms are in fact degenerate trapeziums. Some definitions say a trapezium has only one set of parallel lines, others say a set of parallel lines. I’d have to say that if you’re going to accept squares as rectangles and equilateral isosceles triangles then… welcome to the party!

*edit: there’s also an odd confusion surrounding trapezoid definitions across the waters. It is nicely summarised here.

You might be wondering about the rectangle thing. Well, a rectangle’s strict definition is


Stop teaching the area of a square as base x height too. It’s (side)2. Then maybe students will understand why we call it squared, and stop worrying when asked to find the perimeter or area of a square when given only one side. Also, the diagonal of a square is not the same length as the sides. Use Pythagoras if you don’t believe me.

#3 The trusty triangle

“There are four types of triangle”

No there aren’t. There are dozens! To be fair I’m just being pedantic. I think we should teach students about all the COOL triangles, not just the simplified classification system.

“All triangles have internal angles that sum to 180 degrees”


…except this one. God bless non-Euclidean Geometry.

Also, I’ve seen teachers incorrectly teach the scalene triangle as “a triangle with none of the properties of the other triangles”.

Prepare to have your mind blown. This is a scalene right-angled triangle:


While we’re at it… This is an isosceles right-angled triangle:


Each type of triangle is not necessarily mutually exclusive. An equilateral triangle is technically a ‘special case’ isosceles triangle. It has two equal sides. It just happens to have another equal side too. The only property needed to fulfill ‘scalene’ is that all sides are different lengths. Angles don’t even get a mention. Although three different angles is implied by three different lengths.

#4 Heptagons and Nonagons

These two shapes get a really rough deal. They’re almost NEVER used at school. I blame the internal angle of a heptagon (a nasty decimal) and, well, the nonagon has no excuse. WordPress is even underlining nonagon as an incorrectly spelt word. Do these shapes a favour and talk about them more! Also, try my heptagon puzzle while you’re here.

#5 Area annoyances

I’ll close with this: Think about how confusing it is for our poor students that we frequently interchange the following when describing volume and area:

base, height, width, length and depth. These are all subjective terms. Be mindful.

Also, wouldn’t it be easier if we just taught students how to find the area of a triangle, then applied that to all shapes. All 2-D shapes after all, are made up of triangles. Even circles, kind of.

5 thoughts on “Complements #3 : Properties of Shapes

  1. Wikipedia: “The area of a parallelogram is twice the area of a triangle created by one of its diagonals”
    Whaaa…?! Not even wrong. Doesn’t even … aw, c’mon!
    Concerning for your description of the Wikipedia quote: It’s not a *definition* — is that how Wikipedia describes it? It’s a list of properties.
    The property you call weird for the midpoints of any quadrilateral becomes obvious if you already know that the bisectors of two sides of a triangle must be parallel to the third side.
    I’m splitting hairs here — but although you are right in that the *definition* of a scalene triangle has nothing to do with angles, it is also true that it is logically equivalent to asserting that the three angles are all different. Unless one accepts degenerate triangles (i.e., with two angles equal to 0).
    FYI, I find your site pretty cool. You like great stuff, in my book. Keep it up.

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