Chapter 9.1 — Introduction to Areas

Rectangles and Squares Need Nothing Extra

A rectangle's area comes straight from its two adjacent sides — length and breadth multiplied together — and its perimeter is twice their sum:

A square is simply a rectangle where length and breadth happen to be equal, so both formulas collapse to a single side length:

Nothing about either shape needs an angle measurement or a second reference line — a tape measure along two sides is all either formula asks for.

A Triangle Is Half of Something Bigger

Take any triangle and draw a second, identical copy of it rotated 180° so the two fit together along one shared side — the result is always a parallelogram with the same base and the same height as the original triangle. That single observation is the entire reason the triangle formula carries a ½ in it:

The "height" here is never a slanted side — it is always the perpendicular distance from the base to the opposite vertex, measured at a right angle to the base itself, regardless of whether the triangle is equilateral, isosceles, or scalene.

The Parallelogram Formula Without the Half

Since a parallelogram is exactly two of those triangles joined together, its area drops the ½ entirely:

Any pair of opposite sides can serve as the "base" — what matters is measuring the perpendicular height to whichever side is chosen, not the length of the slanted adjacent side connecting the two parallel sides.

Splitting Any Quadrilateral Along One Diagonal

Rectangles, squares, and parallelograms are all quadrilaterals with special angle or side relationships, but an ordinary, unrestricted quadrilateral — four sides, no guaranteed right angles, no guaranteed parallel sides — needs a different approach entirely. Draw one diagonal across it, say AC in quadrilateral ABCD, and the shape splits into two triangles: ∆ABC and ∆ACD, sharing the diagonal as a common base. Drop a perpendicular from B onto AC (call its length h₁) and another from D onto AC (call its length h₂):

Area of ∆ABC = ½ × d × h₁, and area of ∆ACD = ½ × d × h₂. Adding the two together and factoring out the shared ½ × d gives a rule that covers every quadrilateral, no matter how irregular:

where d is the length of one diagonal and h₁, h₂ are the perpendicular distances from the other two vertices onto that diagonal.

The Rhombus Skips the Diagonal-and-Offset Step

A rhombus is a special quadrilateral whose two diagonals always cross at right angles and bisect each other — and that extra structure lets its area collapse into just the two diagonal lengths, with no separate perpendicular measurements needed at all:

This is really the general quadrilateral formula in disguise: because the diagonals of a rhombus are perpendicular to each other, half of the second diagonal on each side already plays the role that h₁ and h₂ played above, just measured more conveniently through the shape's own symmetry.

The Trapezium Averages Its Two Parallel Sides

A trapezium has exactly one pair of parallel sides, of lengths a and b, separated by a perpendicular distance h. Its area is the average of those two parallel sides, multiplied by the distance between them:

It's worth noticing that a parallelogram is really just a trapezium where the two parallel sides happen to be equal — set a = b in the trapezium formula and it reduces to ½ × 2a × h = a × h, exactly the parallelogram rule seen a moment ago.

Every Formula, Side by Side

With six shapes now covered, it helps to see all six formulas gathered in one place before moving on to problems that mix and combine them:

Notice the pattern running through the last three rows: the quadrilateral, rhombus, and trapezium formulas are all variations of the same "half times something times something" shape, none of them an isolated rule that needs memorising separately from the rest.

Checking the Quadrilateral Formula Against a Familiar Shape

A useful way to trust a new formula is to test it on a shape whose area you can already find another way. Take a square of side 10 cm and draw one diagonal across it. That diagonal has length d = 10√2 cm, and by the symmetry of a square, the perpendicular offsets from the other two corners onto this diagonal are equal: h₁ = h₂ = 5√2 cm each. Plugging these into the general quadrilateral formula:

That matches side × side = 10 × 10 = 100 sq.cm exactly, confirming the general formula doesn't just work for oddly shaped quadrilaterals — it agrees with the simpler square formula whenever the two happen to overlap on the same shape.

Rhombus Diagonals in a Real Tile

The rhombus formula is easiest to trust with real numbers rather than letters alone. A rhombus-shaped floor tile with diagonals 16 cm and 12 cm has area ½ × 16 × 12 = 96 sq.cm — no angle measurement, no side length, just the two diagonals multiplied and halved. Compare that to trying to find the same area using only the side length: a rhombus with those diagonals has sides of length √(8² + 6²) = √(64+36) = √100 = 10 cm, and finding its area from the side alone would additionally require knowing one of its angles, since a rhombus's area isn't fixed by its side length alone — many different rhombuses share a 10 cm side but have different diagonal lengths, and therefore different areas. The diagonal-based formula avoids that ambiguity entirely.

Carrying These Formulas Forward

Straight-line shapes are only half the picture — circles, rings, and sectors follow a different family of formulas built around π, covered later in this chapter. Before that, Exercise 9.1 puts every formula above to direct use on composite figures built by combining two or more of these shapes at once, exactly the kind of splitting-and-adding strategy that real fields, frames, and floor plans actually require.