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
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
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
- 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.