Why can two shapes have the same perimeter but different areas?

They measure different things. Perimeter is the distance around the outside boundary, while area is the surface covered, measured by counting square units. [S1] A boundary’s total length does not tell you exactly how many unit squares the shape covers.

Try two rectangles made from grid squares:

1. Rectangle A is 2 squares wide and 6 squares long.
2. Rectangle B is 3 squares wide and 5 squares long.

For A, the perimeter is 2 + 6 + 2 + 6 = 16 units, and the area is 2 × 6 = 12 square units. For B, the perimeter is 3 + 5 + 3 + 5 = 16 units, but the area is 3 × 5 = 15 square units. The outside distances match; the numbers of covered squares do not.

Use a separate check for each measurement. For perimeter, trace only the outside edge and count its unit-length segments. Do not count lines between squares inside the shape: shared interior edges are not part of the outside distance. [S1] Record the answer in units, such as 16 units.

For area, count the complete unit squares covered by the shape. You can also count rows and columns for a rectangle: 2 rows of 6 squares give 12 square units. Keep the units visibly different in your notes: perimeter in units; area in square units. If your perimeter check accidentally includes an interior grid line, recount the outside boundary only.

Here is a copyable worked comparison. Treat each grid interval as one unit and each small grid square as one square unit:

• Rectangle A, 2 by 6: perimeter check: 2 + 6 + 2 + 6 = 16 units. Area check: 2 × 6 = 12 square units.
• Rectangle B, 3 by 5: perimeter check: 3 + 5 + 3 + 5 = 16 units. Area check: 3 × 5 = 15 square units.

Conclusion: both outside boundaries total 16 units, but the rectangles cover 12 and 15 square units. This example shows that equal perimeter can go with different areas; it does not mean every pair of shapes with equal perimeter must have different areas. Some can have both measurements equal. Calculate or measure each quantity separately.

For a non-rectangle on a square grid, use the same two checks: trace and add the outside unit-length edges for perimeter, then count the covered unit squares for area. If a shape covers partial squares, follow the measurement method your lesson specifies rather than silently treating each partial square as a whole one.

A clear write-up is: “Shape A has perimeter ___ units and area ___ square units. Shape B has perimeter ___ units and area ___ square units. Their perimeters [match/do not match], while their areas [match/do not match]. I checked perimeter by tracing the outside and area by counting covered squares.” If a count disagrees with your formula, redraw the boundary or mark rows of squares to check which edges or squares were missed.