Area is the amount of flat space a shape covers — think of the paint needed to cover a wall or the turf to lay a lawn. It is measured in square units (cm², m²). Perimeter is the total distance all the way around the edge — the fence around that lawn, the skirting board round a room — measured in plain units (cm, m).
Why this matters: these two ideas price nearly every practical job. Carpet, tiles, paint and grass seed are sold by area; fencing, edging, ribbon and picture-frame moulding are sold by length (perimeter). Order by the wrong one and you either run short or waste money, so telling them apart is a genuinely useful life skill, not just an exam topic.
Area of a rectangle = length × width
Area formulas
Area of a triangle uses the perpendicular height, not a slant side.
Every one of these builds on the rectangle. Learn the four and you can handle most straight-sided shapes:
Rectangle = length × width
Triangle = ½ × base × height
Parallelogram = base × height
Trapezium = ½ × (a + b) × height (a and b are the two parallel sides)
Triangle, base 8 cm, height 5 cm: ½ × 8 × 5 = 20 cm². Trapezium, parallel sides 6 and 10, height 4: ½ × (6 + 10) × 4 = ½ × 16 × 4 = 32 cm².
Trapezium: ½(a + b) × h = ½(6 + 10) × 4 = 32 cm².
Why ½ for a triangle? A triangle is exactly half of the rectangle that would box it in (base × height). Slide the two halves together and they make that rectangle — so you take half. A parallelogram, by the same trick, is a rectangle with a corner slice moved across, which is why it is simply base × height.
Perimeter
Perimeter walks the whole edge: 7 + 4 + 7 + 4 = 22 cm.
Perimeter needs no clever formula — you just add up every side, walking round the edge like measuring a running track. For a rectangle that shortcut is 2 × (length + width), since opposite sides match.
Square of area 49 cm²: the side is √49 = 7 cm, so the perimeter is 4 × 7 = 28 cm.
Shape
Area
Perimeter
Rectangle
l × w
2(l + w)
Triangle
½ × base × height
add all 3 sides
Parallelogram
base × height
add all 4 sides
Trapezium
½(a + b) × height
add all 4 sides
Common mistake: muddling area with perimeter (and their units). Area is in square units (cm²) because it is two lengths multiplied; perimeter is in plain units (cm). Also, always use the perpendicular height for area — the straight-up distance — never a slanted side.
Use the straight-up height (dashed), never the slanted side.
Memory hook: area = space inside (square units, cm²); perimeter = distance around (plain units, cm). Learn the four area formulas and you can build the rest from the rectangle.