Area of Parallelograms and Trapeziums

From rectangle to parallelogram

A parallelogram is a four-sided shape with two pairs of parallel sides — like a rectangle that has been pushed over to one side.

Here is the clever part. Imagine slicing a right-angled triangle off the slanted left end of a parallelogram and sliding it across to fill the slanted right end. You are left with a perfect rectangle of the same base and height. Nothing was added or removed, so the area is unchanged:

Area of a parallelogram = base × perpendicular height

Described diagram: picture a leaning rectangle. The bottom side (the base) is flat and 7 cm long. A dashed vertical line goes straight up from the base to the top side, meeting both at right angles; this is the height, say 4 cm. The two sloping sides are longer than 4 cm — ignore them.

Worked example: parallelogram

A parallelogram has base 7 cm and perpendicular height 4 cm.

• Area = base × height = 7 × 4 = 28 cm²

Notice the slanted side length is never used. Only the perpendicular height counts.

The trapezium

A trapezium has exactly one pair of parallel sides. They are usually different lengths — call the longer one b and the shorter one a. The distance between them, measured at a right angle, is the height h.

Area of a trapezium = ½ × (a + b) × height

Why this works

If you take two identical trapeziums and rotate one of them 180°, they fit together to make a parallelogram. That big parallelogram has a base of (a + b) and the same height h, so its area is (a + b) × h. Your single trapezium is half of that, which gives the ½.

Worked example: trapezium

A trapezium has parallel sides of 6 cm and 10 cm, with a perpendicular height of 5 cm.

• Add the parallel sides: 6 + 10 = 16
• Halve it: ½ × 16 = 8
• Multiply by the height: 8 × 5 = 40 cm²

A neat way to remember it: find the average of the parallel sides, then multiply by the height. The average of 6 and 10 is 8, and 8 × 5 = 40.

Working backwards

A parallelogram has area 36 cm² and base 9 cm. Find the height.

• 36 = 9 × height
• height = 36 ÷ 9 = 4 cm

One family of shapes

Shape	Parallel sides	Area formula
Rectangle	both pairs, sides at right angles	length × width
Parallelogram	both pairs	base × height
Trapezium	one pair	½ × (a + b) × height

A rectangle is just a parallelogram with right angles, and a triangle is what you get if one parallel side of a trapezium shrinks to zero. They all share the same underlying idea: average width × height.

Activity: prove it with paper

Cut out a parallelogram from card. Snip a right-angled triangle off one slanted end and slide it to the other end — you will see it become a rectangle. Measure the rectangle's sides and check that base × height gives the area you expect.

Where this leads

These formulas extend the ideas in area and perimeter and connect closely to the area of a triangle, since a triangle is half a parallelogram. Knowing the types of quadrilaterals helps you pick the right formula instantly.