Same area, different perimeter

Mathematics · Measurement · ages 10-11

Name ______________________ Date ____________

Two rectangles both have a perimeter of 20 cm. Can they have different areas?
- Yes, for example a 5 cm by 5 cm square and a 2 cm by 8 cm rectangle
- No, matching perimeters always mean matching areas
- No, only squares can have a perimeter of 20 cm
True or false: two rectangles can have the exact same area but different perimeters.
Circle one: True False
Another rectangle also has area 24 cm2, but it is 12 cm by 2 cm. What is its perimeter, in cm?
Answer: ______________
A rectangle is 4 cm by 6 cm. A second rectangle is 2 cm by 12 cm. Both cover an area of 24 cm². Do the two rectangles have the same perimeter?
- Yes, both perimeters are 20 cm
- No, the perimeters are 20 cm and 28 cm
- No, you cannot find the perimeter without a picture
A long thin rectangle has area 16 cm2 and is 16 cm long. What is its width, in cm?
Answer: ______________
Another rectangle also has perimeter 20 cm, but its length is 8 cm. What is its width, in cm?
Answer: ______________
A fence encloses a rectangular garden with a perimeter of 20 m. Which of these could be the garden's dimensions?
- 3 m by 7 m
- 4 m by 5 m
- 6 m by 6 m
Two rectangles both have perimeter 20 cm. One is 6 cm by 4 cm, the other is 8 cm by 2 cm. Which one has the bigger area?
- 6 cm by 4 cm
- 8 cm by 2 cm
- They have the same area
For a fixed area, a rectangle shaped more like a square has a smaller perimeter than a long, thin rectangle with the same area.
Circle one: True False
Lina says: 'Two rectangles both have a perimeter of 20 cm, so they must cover the same area.' What is wrong with her thinking?
- She measured one of the perimeters wrong
- Two rectangles can never share the same perimeter
- Perimeter measures the distance around a shape, not the space inside it, so matching perimeters don't guarantee matching areas

Answer key

For grown-ups. Fold this page away before handing over the rest.

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Yes, for example a 5 cm by 5 cm square and a 2 cm by 8 cm rectangle · The square has area 25 cm² and the rectangle has area 16 cm². Same perimeter, different area.
True · A 6 by 4 rectangle and a 12 by 2 rectangle both have area 24 cm2, but their perimeters are 20 cm and 28 cm. Area and perimeter don't have to move together.
28 · Even though this rectangle has the same area as a 6 by 4 rectangle, its perimeter is different because the shape is longer and thinner.
No, the perimeters are 20 cm and 28 cm · The first rectangle's perimeter is 2 x (4 + 6) = 20 cm. The second is 2 x (2 + 12) = 28 cm. Same area, different perimeter.
1 · When you know the area and one side, divide to find the missing side.
2 · Same perimeter, different length, so the width has to change too.
3 m by 7 m · The two side lengths must add to 10 m, since 2 x 10 = 20. Only 3 + 7 = 10.
6 cm by 4 cm · Same perimeter does not mean same area. For two rectangles with the same perimeter, the one closer to a square always packs in more area.
True · Squarish shapes pack the same area into a shorter boundary. Stretching a shape out while keeping the area the same makes the perimeter grow.
Perimeter measures the distance around a shape, not the space inside it, so matching perimeters don't guarantee matching areas · A 5-by-5 square (perimeter 20 cm, area 25 cm²) and a 2-by-8 rectangle (perimeter 20 cm, area 16 cm²) both have perimeter 20 cm but different areas.