The formulas, and where they come from
Every area formula on the syllabus reduces to the rectangle. A parallelogram is a rectangle with a triangle moved from one end to the other, so its area is still base times height. A triangle is half a parallelogram. A trapezium is the average of its two parallel sides times the height.
The height in all of these is the perpendicular height, not the slanted side. That distinction is where most marks are lost, because a diagram usually shows the slant length as well and it is the more obvious number.
Perpendicular height, not the slant
In a triangle with sides 5, 6 and 7, the "height" is none of those numbers — it is the perpendicular distance from a vertex to the opposite side. When a diagram gives a slanted side and a dashed perpendicular, the dashed one is the height. If only the three sides are known, use ½ab sin C instead.
Compound shapes
Real questions rarely give a plain rectangle. A compound shape is handled in one of two ways: split it into standard shapes and add the areas, or surround it with a larger standard shape and subtract what is not wanted.
Both work; choose whichever gives fewer pieces. A rectangle with a semicircular bite taken out is one subtraction, whereas splitting it would take three pieces.
Perimeter needs more care than area. When two shapes are joined, the shared edge disappears from the perimeter but the areas still add. And the perimeter of a semicircular section is half the circumference plus the diameter across the flat side — forgetting that straight edge is the standard error.
A running track is a rectangle 100 m by 60 m with a semicircle on each of the two shorter ends. Find its area and perimeter. Take π = 3.142.
- The two semicircles together make one full circle of diameter 60 m, so radius 30 m.Combining the two halves avoids doing the same calculation twice.
- Area
= (100 × 60) + π(30)² = 6000 + 2827.8 = 8827.8m².Rectangle plus circle. - For the perimeter, the two 60 m ends are not part of the boundary — the semicircles replace them.This is where the marks are. Only the two long sides remain straight.
- Perimeter
= (2 × 100) + 2π(30) = 200 + 188.5 = 388.5m.Two straight sides plus one full circumference from the two semicircles.
Area 8827.8 m²; perimeter 388.5 m
Units, and the trap in converting areas
Converting lengths is straightforward: 1 m is 100 cm. Converting areas is where it goes wrong, because the conversion factor must itself be squared.
1 m² is not 100 cm². It is a square one metre on each side, which is 100 cm by 100 cm — 10 000 cm². Similarly 1 m³ is 1 000 000 cm³. The rule follows the same k, k², k³ pattern as similar figures, and for the same reason.
| Length | Area | Volume |
|---|---|---|
| 1 cm = 10 mm | 1 cm² = 100 mm² | 1 cm³ = 1000 mm³ |
| 1 m = 100 cm | 1 m² = 10 000 cm² | 1 m³ = 1 000 000 cm³ |
| 1 km = 1000 m | 1 km² = 1 000 000 m² | — |
| — | 1 hectare = 10 000 m² | 1 litre = 1000 cm³ |
The safest way to convert
Convert the lengths to a common unit before calculating the area, not afterwards. A rectangle 2 m by 30 cm becomes 200 cm by 30 cm, giving 6000 cm² directly — no area conversion needed, and no chance of forgetting to square the factor.
Before you leave this chapter
- Every area formula reduces to base × height; the height is always perpendicular.
- Triangle ½bh, trapezium ½(a+b)h, circle πr² and circumference 2πr.
- Compound shapes: split and add, or surround and subtract — whichever needs fewer pieces.
- A shared edge vanishes from the perimeter but the areas still add.
- 1 m² = 10 000 cm². Convert the lengths first and the problem disappears.
Where the formulas come from
Four area formulas look like four things to memorise. They are one thing seen four ways, and seeing the rearrangement once makes them much harder to confuse — particularly the recurring question of which length is the height.
On the parallelogram, the dashed triangle cut from one end fills the gap at the other exactly. That is why the area is b × h with the perpendicular height — the sloping side never enters into it.
The trapezium contains the others
Set the two parallel sides of a trapezium equal and ½(a + b)h becomes bh — a parallelogram. Shrink one of them to zero and it becomes ½bh — a triangle. One formula covers all three, which is a useful check: if your trapezium answer does not reduce correctly in those two cases, you have misremembered it.
Arc length and sector area
A sector is a fraction of a whole circle, and which fraction is decided entirely by its angle. An angle of θ degrees is θ/360 of a full turn, so the sector has that same fraction of the circumference and of the area.
That single idea replaces two formulas. There is nothing separate to remember: work out the fraction, then take that fraction of whichever whole-circle quantity the question wants.
A sector has radius 10 cm and angle 72°. Find its arc length, its area, and its perimeter. Take π = 3.142.
- Fraction of the circle:
72/360 = 1/5.Simplify it first — every part of the question uses the same fraction. - Circumference
= 2 × 3.142 × 10 = 62.84, so the arc is62.84 ÷ 5 = 12.57cm. - Whole area
= 3.142 × 100 = 314.2, so the sector is314.2 ÷ 5 = 62.84cm². - Perimeter
= arc + two radii = 12.57 + 10 + 10 = 32.57cm.The two straight radii are part of the boundary, and leaving them out is the standard error.
Arc 12.57 cm; area 62.84 cm²; perimeter 32.57 cm
Arc length is not the perimeter
The arc is the curved part alone. The perimeter of the sector is that arc plus the two radii bounding it. Questions ask for one or the other quite deliberately, and giving the arc when the perimeter was wanted loses the final mark every time.