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Geometry & measures · Perimeter, area & volume

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Area of triangles, parallelograms & trapezia

Why every straight-sided area formula is a rectangle in disguise — base times PERPENDICULAR height, the trapezium average, compound shapes by splitting or subtracting, and why 1 m² is 10 000 cm².

Geometry & measures · Perimeter, area & volume

Area of triangles, parallelograms & trapezia

Why every straight-sided area formula is a rectangle in disguise — base times PERPENDICULAR height, the trapezium average, compound shapes by splitting or subtracting, and why 1 m² is 10 000 cm².

Why it works

Area counts unit squares, and a rectangle counts them instantly: length×width\text{length} \times \text{width} rows of squares. Every other straight-sided formula comes from turning the shape into a rectangle.

Parallelogram: shear it square. Slice the right-angled triangle off one end of a parallelogram and slide it round to the other end — you get a rectangle with the same base and the same vertical height. So

A=base×perpendicular height.A = \text{base} \times \text{perpendicular height}.

The slant side is not the height. Exam figures deliberately print both — the perpendicular height (often dashed, with a right-angle mark) and the longer slant side. Multiplying base by slant is the classic lost mark.

Triangle: half a parallelogram. Two copies of any triangle fit together into a parallelogram, so A=12×base×perpendicular heightA = \frac{1}{2} \times \text{base} \times \text{perpendicular height}. Any side can be the base — but the height must be measured at right angles to that base.

Trapezium: average the parallel sides. Two copies of a trapezium (one rotated) make a parallelogram of base a+ba + b and the same height, so

A=(a+b)2×hA = \frac{(a + b)}{2} \times h

— the average of the two parallel sides, times the perpendicular distance between them. Adding without halving, or using a slant leg as hh, are the two standard slips.

Compound shapes: split or subtract — then label. Cut an L-shape into two rectangles, or see it as a big rectangle minus a bite. Before any arithmetic, work out the missing side lengths from the ones given (opposite sides of the full rectangle must match up). Perimeter is a different question: it adds lengths around the outside and never multiplies.

Area units square the conversion. 11 m =100= 100 cm, but a square metre is a 100×100100 \times 100 grid of centimetre squares: 1 m2=10 000 cm21 \text{ m}^2 = 10\,000 \text{ cm}^2. Converting area with a factor of 100100 is a very common error — the factor gets squared because the units are.