Same area, different perimeter
When figures have the same area but different perimeters, the key is to keep track of what is fixed and what changes. Two shapes can cover the same amount of space while using different lengths around the outside.
1) Find or confirm the area
- Use the given measurements to calculate the area.
- If the exercise asks for a figure with the same area, make sure your dimensions produce that same value.
2) Compare the perimeters
- Perimeter is the total distance around the outside.
- Even when the area stays the same, changing side lengths can change the perimeter.
- For rectangles, for example, a long thin shape and a more square-like shape can have the same area but different perimeters.
3) Simplify the final answer
- If your answer is an expression, combine like terms and reduce fractions if needed.
- If you are choosing between shapes, state the perimeter clearly and compare it to the others.
4) Check your work
- Recompute the area to confirm it matches the target.
- Recompute the perimeter from the final dimensions.
- Make sure the answer is exact if exact values are required.
A good strategy is to test dimensions systematically: keep the area constant first, then see how the perimeter changes.