calculate the longest and the shortest perimeter possible that encloses and area of 9m..16m..25m..36.. can u tell me please
To calculate the longest and shortest perimeter possible that encloses an area, we need to consider all possible shapes with the given areas and find the minimum and maximum perimeters for those shapes.
Let's start with the given areas:
1. Area = 9m²
The shape with the smallest area that can enclose 9m² is a square. Since all sides of a square are equal, the perimeter of the square can be calculated by adding all four sides together: 4 * side length. Therefore, the shortest perimeter for an area of 9m² would be the perimeter of a square with sides of length √9m = 3m.
The longest perimeter for an area of 9m² can be found by considering different shapes that can enclose this area. In this case, a square with sides of length 3m already provides the maximum perimeter.
2. Area = 16m²
The only shape that can enclose 16m² is again a square. Using the same logic as above, the shortest perimeter for an area of 16m² would be 4 * √16m = 4 * 4m = 16m.
Similarly, the longest perimeter for an area of 16m² can be obtained using a square with sides of length 4m, which also gives a perimeter of 16m.
3. Area = 25m²
Once again, the shape with the minimum area enclosing 25m² is a square. The shortest perimeter would be 4 * √25m = 4 * 5m = 20m.
The longest perimeter for an area of 25m² can be achieved using a shape with circular symmetry, which is a circle. In this case, we need to calculate the circumference of a circle with an area of 25m². The formula for the circumference of a circle is 2πr, where r is the radius of the circle. Solving for the radius, we get r = √(25m²/π) ≈ 2.83m. Therefore, the longest perimeter would be 2π * 2.83m ≈ 17.8m.
4. Area = 36m²
The shape with the smallest area enclosing 36m² is a square. The shortest perimeter would be 4 * √36m = 4 * 6m = 24m.
Similar to the previous case, the longest perimeter for an area of 36m² can be achieved using a circle. The radius of the circle can be calculated as r = √(36m²/π) ≈ 3.82m. Therefore, the longest perimeter would be 2π * 3.82m ≈ 24m.
In summary:
- For an area of 9m², the shortest and longest perimeters are 3m and 12m respectively.
- For an area of 16m², the shortest and longest perimeters are 16m.
- For an area of 25m², the shortest perimeter is 20m, and the longest perimeter is approximately 17.8m.
- For an area of 36m², the shortest and longest perimeters are 24m.