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SSG-E
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- For example, Consider two shapes; a circle and rectangle.
Both these shapes have same area but the perimeter of circle is less than that of rectangle. Why?
SSG-E said:Summary:: For example, Consider two shapes; a circle and rectangle.
Both these shapes have same area but the perimeter of circle is less than that of rectangle. Why?
nonePeroK said:Or, try to find an example of two different shapes that have the same area and the same perimeter.
I mean is it something related to the sides of the shapes?phyzguy said:I don't know what kind of answer you are looking for. It's obvious from geometry that there are many possible shapes with the same area and different perimeters. If I make a very long skinny rectangle of a given area, I can make the perimeter as large as I want. I don't know what kind of answer to give to your question except, "that's the way the geometry of our universe works."
The answer is trivially yes.SSG-E said:I mean is it something related to the sides of the shapes?
yesDrClaude said:The answer is trivially yes.
Start with a simple problem: consider a rectangle with sides of length ##a## and ##b##. The area is given by ##a \times b## while the parameter is ##2 (a+b)##. Do you see how a quantity that changes with the product of variables behaves differently than a quantity that changes with the sum of those variables?
There are examples, but they are a bit harder to find. A triangle and a suitable trapezoid are the easiest examples.SSG-E said:nonePeroK said:Or, try to find an example of two different shapes that have the same area and the same perimeter.
triangles in some cases?mfb said:There are examples, but they are a bit harder to find. A triangle and a suitable trapezoid are the easiest examples.
PeroK said:Or, try to find an example of two different shapes that have the same area and the same perimeter.
Circle maximizes Area/perimeter. Sphere has the largest Volume/area.etotheipi said:Another exercise would be to try and prove which shape maximises ##\frac{\text{Area}}{\text{Perimeter}}##. And what about ##\frac{\text{Volume}}{\text{Surface Area}}## in 3D?
SSG-E said:Circle maximizes Area/perimeter. Sphere has the largest Volume/area.
In case of circle, it has infinite number of sides. In case of a sphere, For example, balloons are spherical, and they will assume the shape of minimum areaetotheipi said:It's right, but can you prove it?
SSG-E said:For example, balloons are spherical, and they will assume the shape of minimum area
Thinking about it, two suitable trapezoids are a much easier example of two different shapes with the same area and perimeter.mfb said:There are examples, but they are a bit harder to find. A triangle and a suitable trapezoid are the easiest examples.
I have in mind flopping between trapezoid to a parallelogram.mfb said:Thinking about it, two suitable trapezoids are a much easier example of two different shapes with the same area and perimeter.
etotheipi said:For a more mathematical approach, you might look at something like this.
The perimeter of a shape is the distance around its outer edge. The area of a shape is the measure of the space inside its boundaries. So, while two shapes may have the same area, their perimeters can be different because they have different boundary lengths.
Yes, consider a square and a rectangle with the same area of 16 square units. The square has a perimeter of 16 units (4 sides x 4 units per side), while the rectangle could have a perimeter of 20 units (2 sides x 8 units per side + 2 sides x 2 units per side).
The number of sides of a shape directly affects its perimeter. The more sides a shape has, the longer its perimeter will be for a given area. This is because each additional side adds to the total boundary length of the shape.
Yes, there are some special cases where shapes with the same area can have the same perimeter. For example, two circles with the same area will have the same perimeter, as the circumference of a circle is directly proportional to its radius.
Understanding the relationship between area and perimeter can help us solve real-world problems involving shapes. For example, if we know the area and perimeter of a field, we can calculate the length of fencing needed to enclose it. It also helps us compare and classify shapes based on their properties.