I am looking for a good example on why areas of the same m^ do not have the same circumference and cannot find a way to put it into words.
Example.
12m x 8m = 96m (12 +12 +8 + 8 = 40 metres of perimeter)
sqrt 96m = 9.7979m rounded.
9.7979m x 4 = 39.1918m rounded of perimeter.
Where did the 0.8m go?
stan101 said:
I am looking for a good example on why areas of the same m^ do not have the same circumference and cannot find a way to put it into words.Example.
12m x 8m = 96m (12 +12 +8 + 8 = 40 metres of perimeter)
sqrt 96m = 9.7979m rounded.
9.7979m x 4 = 39.1918m rounded of perimeter.Where did the 0.8m go?
Sqrt 96m is an irrational number, so squaring it won’t give you a “whole number” answer.
>The square roots of numbers that are not a perfect square are members of the irrational numbers. This means that they can’t be written as the quotient of two integers. The decimal form of an irrational number will neither terminate nor repeat. The irrational numbers together with the rational numbers constitutes the real numbers.
https://www.mathplanet.com/education/pre-algebra/right-triangles-and-algebra/square-roots-and-real-numbers
stan101 said:
I am looking for a good example on why areas of the same m^ do not have the same circumference and cannot find a way to put it into words.Example.
12m x 8m = 96m (12 +12 +8 + 8 = 40 metres of perimeter)
sqrt 96m = 9.7979m rounded.
9.7979m x 4 = 39.1918m rounded of perimeter.Where did the 0.8m go?
One way to look at it is to divide the area into a number of equal squares. All the squares around the edge will contribute at least 1, and up to 3, of their edges to the perimeter length. To minimise the perimeter you need to make as many squares as possible internal, and equalising the width and length allows you to do that.
http://www.byrdseed.com/explore-geometry-area-and-perimeter/
stan101 said:
I am looking for a good example on why areas of the same m^ do not have the same circumference and cannot find a way to put it into words.Example.
12m x 8m = 96m (12 +12 +8 + 8 = 40 metres of perimeter)
sqrt 96m = 9.7979m rounded.
9.7979m x 4 = 39.1918m rounded of perimeter.Where did the 0.8m go?
It did not “go’ anywhere. That is just the perimeter for that size square.
Let’s say you made the rectangle 24 m x 4 m instead. The area is the same at 96 m^2 but the perimeter will be far greater, 24 +24 + 4 + 4 = 56 m.
Perimeter length is only relative to area if the shape is the same.
Shapes with perimeters of 100m have areas of:
Circle – 795m2
Square – 625m2
Rectangle (40×10) – 400m2
Rectangle (30×20) – 600m2
diddly-squat said:
http://www.byrdseed.com/explore-geometry-area-and-perimeter/
Thanks everyone for the explanations. I think the most basic way to demonstrate it might be to show the table in the above link.
Length Width Perimeter Area
1 m 9 m 20 m 9 sq m
2 m 8 m 20 m 16 sq m
3 m 7 m 20 m 21 sq m
4 m 6 m 20 m 24 sq m
5 m 5 m 20 m 25 sq m
I might draw those shapes and dimension them as well.
Cheers,
The Rev Dodgson said:
One way to look at it is to divide the area into a number of equal squares. All the squares around the edge will contribute at least 1, and up to 3, of their edges to the perimeter length. To minimise the perimeter you need to make as many squares as possible internal, and equalising the width and length allows you to do that.
This might work with cardboard cutout of the main quadrilateral and then some smaller even sized ones. That way it can be easily visualised.
Thanks again all. My google search didn’t bring up info this good
Remember, a circle wil have the greatest area for its perimeter.
The smallest area is likely formed by an extremely long and thin rectangle, or maybe by something like this…
Speedy said:
Remember, a circle wil have the greatest area for its perimeter.The smallest area is likely formed by an extremely long and thin rectangle, or maybe by something like this…
There is a shape with an infinite perimeter and finite area.
In fact there are infinitely many such shapes, but the one I’m thinking of has a special name, which I have forgotten.
https://en.wikipedia.org/wiki/Koch_snowflake
stan101 said:
This might work with cardboard cutout of the main quadrilateral and then some smaller even sized ones. That way it can be easily visualised.
We invested in a set of these wooden blocks when the boys were in their first year of school. Now they are both in high school and the blocks are still used on occasion for multiplication, area and volume.
We have tried to squeeze the same blocks that fit into the squares into rectangles of the same perimeter, but they don’t all fit.
The Rev Dodgson said:
https://en.wikipedia.org/wiki/Koch_snowflake
Arghh…Koch curve… Make it stop!
I know the possibilities are endless, but stan101 could have some fun explaining this and watching the process of thought. I wonder what shapes his “pupils” will discover themselves :)
to make it even tougher, my “students” don’t have english as their mother-tongue and some lack any english at all. They are all actually well more educated than I am. I am just trying to demonstrate some real life scenarios.
I try to break things down and work with just basic visual concepts. It can be interesting, fun, entertaining and frustrating for all involved at times :)
Am I assuming the Koch Curve and a fractal are one in the same?
stan101 said:
to make it even tougher, my “students” don’t have english as their mother-tongue and some lack any english at all. They are all actually well more educated than I am. I am just trying to demonstrate some real life scenarios.I try to break things down and work with just basic visual concepts. It can be interesting, fun, entertaining and frustrating for all involved at times :)
Am I assuming the Koch Curve and a fractal are one in the same?
It’s an example of a fractal. There are many more complicated ones.
Speedy said:
Remember, a circle wil have the greatest area for its perimeter.The smallest area is likely formed by an extremely long and thin rectangle, or maybe by something like this…
Indeed a finite area can have an infinite perimeter
