Two different measures
Perimeter measures the distance around a shape (a length, in cm). Area measures the flat space inside a shape (in cm²). They are not the same, and one does not decide the other.
Key idea / 要点
Shapes with the same perimeter can have different areas, and shapes with the same area can have different perimeters.
Same perimeter, different area
Worked example / 例题
Two rectangles each have perimeter 20 cm.
Rectangle A: 6 cm × 4 cm → area = 6 × 4 = 24 cm².
Rectangle B: 8 cm × 2 cm → area = 8 × 2 = 16 cm².
Same perimeter (20 cm) but different areas. For a fixed perimeter, the area is largest when the rectangle is closest to a square.
Same area, different perimeter
Two rectangles can both have area 24 cm²: a 6 × 4 rectangle has perimeter 2 × (6 + 4) = 20 cm, while a 12 × 2 rectangle has perimeter 2 × (12 + 2) = 28 cm. Same area, different perimeters. This is useful in real life: a farmer wanting the most grazing land from a fixed length of fence should make the field as square as possible, while a builder saving on wall length for a fixed floor area should also avoid long, thin rooms.
So whenever a question changes the shape of a figure, check both measures separately. Never assume that because one shape has a bigger perimeter it must also have a bigger area — work each one out from its own formula.
Remember / 记住
- Perimeter is in cm; area is in cm² — they are different quantities.
- Equal perimeter does not force equal area (and vice versa).
- For a fixed perimeter, a square-like shape gives the biggest area.