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Fraction Classroom · 2.3.0 · Student pages

Keep the boundary length and change the area

Compare two rectangles with the same perimeter and use unit squares to show why their areas can differ.

Anonymous group code: __________   Date: __________

Each small square is 1 square cm. Both rectangles have perimeter 16 cm; the first covers 15 squares and the second 12.5 cm × 3 cm6 cm × 2 cm
Each small square is 1 square cm. Both rectangles have perimeter 16 cm; the first covers 15 squares and the second 12.

Build the idea

model. A 5 cm by 3 cm rectangle and a 6 cm by 2 cm rectangle both need 16 cm of edging. Do they cover equal areas?

  1. Trace the entire outside of the first rectangle. Add all four side lengths, including the opposite sides.
  2. Tile its inside with square-centimetre units. Count five in each of three rows to find 15 square centimetres.
  3. Repeat both measurements for the 6-by-2 rectangle. Its boundary is still 16 cm, but only 12 unit squares fit inside.
  4. State what the example proves: equal perimeters do not force equal areas. Keep cm for boundary length and square cm for covered area.

Worked example: No. Their perimeters are 5 + 3 + 5 + 3 = 16 cm and 6 + 2 + 6 + 2 = 16 cm. Their areas are 15 and 12 square centimetres.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. Find the perimeter in cm of a rectangle 7 cm long and 2 cm wide.

p2. Find the area in square cm of that 7 cm by 2 cm rectangle.

p3. A 6 cm by 3 cm rectangle also has perimeter 18 cm. Find its area in square cm.

e1. A 4 cm by 4 cm square and a 7 cm by 1 cm rectangle both have perimeter 16 cm. How many more square cm does the square cover?

Transfer the idea

transfer. Can two rectangles have the same area but different perimeters? Give dimensions and calculate both measurements.