Calculating the area of a circle or square using decimals

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Discussion Overview

The discussion revolves around the calculation of area for squares and rectangles, particularly focusing on the implications of using decimal measurements. Participants explore the counter-intuitive nature of areas being smaller than the lengths of their sides and the proper use of units in these calculations.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants express confusion about how an area can be smaller than the length of a side when using decimal measurements, specifically citing a square with a side of 0.8m resulting in an area of 0.64m².
  • Others clarify that the area calculation is correct, emphasizing the importance of using proper units, noting that 0.8m × 0.8m equals 0.64m², and that 80cm × 80cm equals 6400cm², which is also 0.64m².
  • A participant suggests visualizing the problem by drawing squares to better understand the relationship between side lengths and area.
  • Some participants challenge the validity of the original question regarding how an area can be smaller than a side, comparing it to other unrelated measurements, such as age and height.
  • Another participant proposes sketching a graph of y = x² to illustrate differences in areas for values of x between 0 to 1 and 1 to 3.
  • One participant acknowledges their misunderstanding after visualizing the problem, indicating a shift in their comprehension of the concept.
  • A participant notes that reducing one dimension of a rectangle from 1.0 to 0.8 results in a smaller area, reinforcing the relationship between dimensions and area.

Areas of Agreement / Disagreement

Participants generally agree on the mathematical principles involved in calculating area and the importance of using correct units. However, there remains some disagreement regarding the framing of the initial question and the implications of the area being smaller than the side length.

Contextual Notes

Some limitations in understanding arise from the use of incorrect units and the framing of questions, which may lead to confusion about the relationships between different mathematical concepts.

Camdenl
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I came across something that is completely counter-intuitive, and I'm wondering if I'm correct or not. If a square has a side that is .8m someone would do .8 time .8 which is .64. How can an area be smaller than a side I thought and so I looked it up and found only one site that said something that made sense. They said that if it was smaller than 1 then you should scale down the measurement so that it would be a whole number. So I tried 80cm by 80cm which came out to 6400cm which is 64m. When you multiply .8 meters by .8 meters it comes out to .64 meters. Am I correct, and if so, why isn't this taught in schools?
 
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Camdenl said:
So I tried 80cm by 80cm which came out to 6400cm which is 64m

How many square centimetres are there in a square metre ?

What is the correct abbreviated way to write 'square metres' - is it m or is it m2 ?
 
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Camdenl said:
I came across something that is completely counter-intuitive, and I'm wondering if I'm correct or not. If a square has a side that is .8m someone would do .8 time .8 which is .64. How can an area be smaller than a side I thought and so I looked it up and found only one site that said something that made sense. They said that if it was smaller than 1 then you should scale down the measurement so that it would be a whole number. So I tried 80cm by 80cm which came out to 6400cm which is 64m. When you multiply .8 meters by .8 meters it comes out to .64 meters. Am I correct, and if so, why isn't this taught in schools?
Careful with the units! That's 0.8 m × 0.8 m = 0.64 m2. Likewise, 80 cm × 80 cm = 6400 cm2 = 0.64 m2.

To visualize it, draw a square of side 1 and another square of size 0.8 inside it (make one the corners to coincide). You will see how the resulting small square is about 64% the size of the big square.
 
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Nidum said:
How many square centimetres are there in a square metre ?

What is the correct abbreviated way to write 'square metres' - is it m or is it m2 ?
I understand that I didn't use the correct way to show measurements, but that still doesn't help my question. Your answer is the equivalent of me asking "Are there things that shouldn't be an adverb." and you respond with "You forgot an apostrophe in 'shouldn't'."
 
Camdenl said:
I understand that I didn't use the correct way to show measurements, but that still doesn't help my question. Your answer is the equivalent of me asking "Are there things that shouldn't be an adverb." and you respond with "You forgot an apostrophe in 'shouldn't'."
Actually, it is crucial to your question. You wrote:
Camdenl said:
How can an area be smaller than a side
which you will figure out is not a valid question with the use of correct units.
 
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Since the units are different, as others have pointed out, they are different things. Your question of , "How can an area be smaller than a side?" is like asking, "How can my age be less than my height?" or "How can 1 tree be less than 100 apples when all of the apples are growing on the tree?"
 
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Forget the area problem itself and just think about how numbers work .

Sketch a graph of y = x2 for x = 0 to 3 . What is different about the sections of the graph for where x is in the range 0 to 1 and for where x is in the range 1 to 3 ?
 
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DrClaude said:
Careful with the units! That's 0.8 m × 0.8 m = 0.64 m2. Likewise, 80 cm × 80 cm = 6400 cm2 = 0.64 m2.

To visualize it, draw a square of side 1 and another square of size 0.8 inside it (make one the corners to coincide). You will see how the resulting small square is about 64% the size of the big square.
Sorry for being rude, I fully understand now. When I visualized i thought of a square with .5cm sides and then a square that is 1cm^2. When I visualized putting the square in the .5 by .5 I realized the area only took up 1/4 of the square.
 
Nidum said:
Forget the area problem itself and just think about how numbers work .

Sketch a graph of y = x2 for x = 0 to 3 . What is different about the sections of the graph for where x is in the range 0 to 1 and for where x is in the range 1 to 3 ?
Thank you for helping me understand the concept, it really helped.
 
  • #10
A rectangle of dimensions 0.8 by 1.0 will have an area of 0.8. When the second dimension is reduced from 1.0 to 0.8, the area gets smaller.
 

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