L3 · Area versus Perimeter: Same Area, Different Perimeter

Learning goal

“I can show that rectangles with the SAME area can have DIFFERENT perimeters, find the perimeters of several rectangles with a given area, and state the rule: among all rectangles with the same area, the square has the least perimeter.”

Dimensions — the pair of side lengths that describe a rectangle, written like "6 cm by 4 cm". Changing the dimensions changes the shape even when the area stays the same.

Factor pair — two whole numbers that multiply to a given number. The factor pairs of 24 are 1 x 24, 2 x 12, 3 x 8, 4 x 6 - each pair is a possible set of dimensions for a rectangle of area 24.

Least perimeter — the smallest possible perimeter for a given area. It belongs to the square (or the rectangle closest to a square).

Worked example

Find rectangles with area 24 cm2, and their perimeters. Step 1: The dimensions must be factor pairs of 24: 1 x 24, 2 x 12, 3 x 8, 4 x 6. Step 2: Perimeter of each: 1 x 24 -> 1 + 24 + 1 + 24 = 50 cm. 2 x 12 -> 28 cm. 3 x 8 -> 22 cm. 4 x 6 -> 20 cm. Step 3: Read the table: the AREA never changed (24 cm2 every time) but the PERIMETER took four different values: 50, 28, 22, 20 cm. Say it plainly: area does NOT determine perimeter. A long skinny rectangle and a compact one can cover exactly the same surface while needing very different fences.

Check yourself — explain it

Two rectangles both have area 24 cm2. Their perimeters...

Paired problem — example still visible

Rectangles with area 12 cm2: dimensions 2 x 6 and 3 x 4. Perimeters? -> 2 x 6: 16 cm; 3 x 4: 14 cm - different perimeters, same area

Faded completion — last step is yours

"Area 18 cm2, dimensions 9 x 2: P = 9 + 2 + 9 + ___ = ___ cm" -> 2; 22 cm

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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