Foundations — Area — triangle, parallelogram, trapezium, composite shapes

This page is the toolbox. Before you touch triangles, parallelograms or trapeziums on the parent topic, you need every symbol and idea it silently assumes. We build each one from nothing, in the order that lets the next one make sense.

1. The unit square — what "area" actually counts

Before any letter or formula, there is a picture: a tiny square whose side is one unit long (1 cm, 1 m, whatever your ruler uses).

Look at the figure. The big rectangle is $4$ wide and $3$ tall. Count the little red squares: there are $4$ in each row and $3$ rows, so $4\times 3=12$ squares fit inside. That is why a rectangle's area is width times height — multiplication is repeated counting of rows.

2. The symbol $A$ and "square units"

Where does that raised $2$ come from? It is the same exponent (repeated-multiply) notation you use for numbers: $3^2=3\times 3$ means "$3$ multiplied by itself twice". A square patch whose side is $1$ cm has area $1\text{ cm}\times 1\text{ cm}$, and we abbreviate that repeated multiplication of the unit as ${\text{cm}}^2$ — read aloud as "centimetre squared". The exponent $2$ literally records that we multiplied a length by a length, i.e. that area spans two directions.

3. Base and height — the two measurements that matter

Now the two words every formula leans on: base and height. From here on the letters $b$ (base) and $h$ (height) have precise meanings — we define them before using them anywhere.

This is the single most important picture on the page. The base is the black bottom line. The red line is the height — notice it goes straight up, meeting the base at a right angle (the small square marks $90°$). The slanted grey side is longer than the red height, and that is exactly the trap: the slant is not the height.

4. Multiplication as "rows of squares" — the operation $\times$

Now that $b$ and $h$ mean something, name the operation that joins them. The symbol $\times$ in an area formula is not abstract — it is "how many rows" times "how many in a row". When a formula writes $b\times h$ it is literally saying lay down $h$ rows, each holding $b$ squares — exactly the counting you did in Section 1.

5. The right angle and the perpendicular symbol

Why does the topic obsess over this? Because area measures straight-up extent. Any measurement that isn't perpendicular to your base is measuring partly sideways, which over-counts. The right-angle mark is your promise that a length is "honest height".

6. The parallelogram area formula — a leaning rectangle

Once $b$ and $h$ are pinned down, the first shape beyond the rectangle almost writes itself.

Why is it the same $bh$ as a rectangle, even though it leans? Slice the leaning triangle off one end and slide it to the other end — you get back a plain rectangle of base $b$ and height $h$. The lean rearranges the tiles but never adds or removes any, so the area is unchanged. Notice it is $b$ times the perpendicular $h$, never the slant side.

7. The fraction $\frac{1}{2}$ — "half of"

If you can see why a triangle is exactly half a rectangle-shaped box around it, you never need to memorise the $\frac{1}{2}$ — you rebuild it.

Two copies of the same triangle (one red, one flipped) snap together into a parallelogram of base $b$ and height $h$. The parallelogram's area is $bh$ (Section 6), so each triangle is therefore $\frac{1}{2}bh$.

8. The trapezium and its area — averaging two parallel sides

Why the average $\frac{a+b}{2}$? Duplicate the trapezium, flip the copy, and glue the two together along a slanted side — they combine into a parallelogram whose base is $a+b$ and height is $h$. That parallelogram has area $(a+b)h$, and your trapezium is half of it, giving $\frac{1}{2}(a+b)h=\frac{a+b}{2}h$. In words: a trapezium behaves like a rectangle whose width is the average of its two parallel sides.

9. Letters as stand-ins — $b$, $h$, $a$

10. Adding and subtracting areas — the $+$ and $-$ for composite shapes

Composite shapes need only two ideas you already own from counting:

The $+$ means "put the tiles together"; the $-$ means "take some tiles away". Nothing deeper.

11. Sine — borrowed only when the height is hidden

Sometimes you're handed the slant side and an angle, not the height. To recover the honest perpendicular height you borrow one tool from trigonometry.

You don't need the full theory yet — just know it exists and answers exactly one question: "how much of this leaning side is actually vertical?" The deep dive lives in 2.1.4-Trigonometric-ratios.

How these feed the topic

Read the map top to bottom: each box below rests on the ones with arrows pointing into it. The rectangle (built from unit squares and multiplication) is the trunk; triangle, parallelogram and trapezium are branches; composite shapes gather them all.

Related building blocks in the vault: 1.2.1-Basic-shapes name the shapes, 1.2.11-Perimeter measures their edges, 1.3.5-Pythagorean-theorem recovers missing lengths, and much later 3.4.2-Integrationas-area pushes this same "count the tiles" idea to curved regions. Circles get their own treatment in 1.2.13-Circle-area-and-circumference.

Equipment checklist

Cover the right side and see if you can answer each before you move on.