Unit 12: Four Sides

Everyone is somebody’s special case

A trapezium has one pair of parallel sides.

A parallelogram has two pairs — so it is already a trapezium.

A rectangle is a parallelogram with right angles. A rhombus is a parallelogram with equal sides. And a square is both at once.

So every square is a rectangle. Not the other way round.

Four angles make a full turn and a half

Draw one diagonal across any four-sided shape and it splits into two triangles.

Each triangle holds 180°, so the four angles must add to 360°.

That is the same trick you will use in unit 13 for shapes with any number of sides.

Parallelograms

In a parallelogram, three things are always true:

- Opposite sides are equal.
- Opposite angles are equal.
- The diagonals cut each other in half.

All three follow from the sides being parallel, and all three are inherited by rectangles, rhombuses and squares.

What the diagonals tell you

The diagonals are where the family really separates.

In a rectangle the two diagonals are equal.
In a rhombus they cross at right angles.
In a square — being both — they are equal *and* cross at right angles.

So you can identify a shape by its diagonals alone.

Perimeter is just adding

The perimeter is the distance all the way round the outside. Add the sides up. That is genuinely all.

For shapes with matching sides there are shortcuts. A rectangle is 2(l + w); a square is 4s; a rhombus is 4s as well, since all four sides are equal.

Perimeter is a length, so it is measured in cm or m — never cm².

Same perimeter, different shape

Two rectangles can share a perimeter and look nothing alike.

With perimeter 20, the sides could be 1 and 9 (area 9), or 4 and 6 (area 24), or 5 and 5 (area 25).

So perimeter tells you almost nothing about area. The square always wins on area — which is the inequality from unit 22, arriving early.

The kite

A kite has two pairs of equal sides — but the equal ones are next to each other, not opposite. That is what makes it a kite rather than a parallelogram.

One pair of opposite angles is equal: the two at the sides, not the ones at the point and the tail.

And its diagonals cross at right angles, with the long one cutting the short one exactly in half.

The isosceles trapezium

A trapezium has one pair of parallel sides. If the other two sides are equal, it is isosceles — and then it gains extra properties.

The two angles along each parallel side are equal, and the diagonals come out the same length.

It is the shape of a lampshade seen from the side, and it has one line of symmetry straight down the middle.

Finding the missing corner

Three corners of a rectangle are at (1, 1), (6, 1) and (1, 4). Where is the fourth?

The bottom side runs from 1 to 6 across. The left side runs from 1 to 4 up. So the last corner is (6, 4).

Opposite sides of a rectangle are equal and parallel, which on a grid means the same jump repeated.

Checking a shape really is what it claims

To prove four points make a square, check two things: all four sides the same length, and the diagonals equal.

Sides equal alone would allow a rhombus. Diagonals equal alone would allow a rectangle.

Both together leave only the square. Coordinates turn "it looks like a square" into something you can actually prove.

Enough, and not more than enough

What is the least you must check to be sure a shape is a parallelogram?

Any one of these will do: both pairs of opposite sides equal, both pairs of opposite angles equal, or the diagonals cutting each other in half.

One is enough. Checking all three is not wrong, just wasted effort — and knowing which single fact suffices is what makes a proof short.

One counterexample kills a claim

"A quadrilateral with four equal sides must be a square." Is that true?

No. A rhombus that has been pushed over has four equal sides and no right angles at all.

That single shape is a counterexample, and it settles the question completely — the tool unit 28 will build a whole lesson on.

A parallelogram is a rectangle in disguise

Slice the sloping end off a parallelogram and slide it round to the other side. You get a rectangle with the same area.

So the area is base × height — and the height means straight up, not along the slanted side.

That sloping side is a trap, and it is put there deliberately in exam questions.

A trapezium: average the parallel sides

A trapezium has two parallel sides of different lengths. Take their average, then multiply by the height.

Area = (a + b)/2 × h

Why the average? Turn a second copy upside down and fit it against the first — together they make a parallelogram with base a + b, and you have exactly half of it.