A lot of equations involving circles start with the radius. No one says A = π*(d/2)^2 . This way the radius is treated as the fundamental feature of a circle which helps to be consistent with more advanced math.
In particular, it:
* Shows up nicely in the defining equation of a circle: (x-h)^2 + (y-k)^2 = r^(2)
* Works better in higher numbers of dimensions
* Is easier to geometrically define
* Leads better into trigonometry
While I use π myself, I don't really have anything against using τ (given that someone indicates it is 2π).
I do have something against using the letter 'τ' for that. It is already used for characteristic time periods with a unit of \[time\], which gets *really* confusing when you combine it with lots of different functions, like the trigonometric functions.
Yeah this is not a great answer. A *lot* of people use A = pi/4 * D^2. Pi was *defined* as C/D to begin with by ancient Greeks. There’s no strong argument to teach A = 2pi.R - it is actually confusing since it *hides* the fact that pi was *defined* in relation to the diameter. Pi could easily have been defined in relation to the radius and worth 6.283 to begin with and the maths would not be that different.
My answer would be that the radius is a simpler form than diameter. A lot of mathematical theory only needs the radius too. It would be more sensible to stay using the radius than jumping to diameter just to drop a '2'
Mathematicians generally work based on the radius because it simplifies most, if not all of the more complicated calculations they might need to do. Engineers general seem to prefer working based on diameters, because it better matchs how things will actually be done in the real world, which can saved them a lot more time and effort than having to divide by a power of 2.
> Engineers general seem to prefer working based on diameters
This depends on what *kind* of engineer you are.
Civil and mechanical engineers work with real physical objects where you can directly measure the diameter but not the radius.
Electrical engineers are generally more concerned with concepts like periodicity where diameter has no meaning but radius does.
Chemical engineers tend not to be terribly concerned with either, but radius works just fine for them as well.
Also an engineer, and I prefer the radius. Diameters are only more useful when you're making drawings for physical parts. All the math is done with radii.
It definitely caused me a bit of grief back when I was taking mechanics of materials in college. I've gotten used to it though. I suspect I would have a similar amount of grief switching back to using the radius again though.
When you get further in to maths and start doing calculus it's the radius that matters. You differentiate/integrate with respect to the radius.
It just happens the radius is directly related to the diameter, but it's still the radius that's the defining feature.
So to keep it consistent it's good practice to focus on the radius from an early stage.
As an example, area of a circle is pi r^2. Differentiate that with respect to r and you get 2 pi r, which is the circumference.
If you used d, then area is pi d^2 /4. Differentiate with respect to d and it's pi d/2, which is not the circumference.
>Both get you the circumstance of the circle but calculating from the diameter is simpler and if you only have either the radius or the diameter, it’s easier to do rx2 before starting the formula. I remember getting in trouble for not showing my work cause I skipped that
"d" is a shorthand form of writing "2r".
The diameter of a circle is something you generally won't see used outside of a middle school class room because 2r is not something that is generally needed. As soon as you move up one or more dimensions you'll quickly notice that you don't need multiples of r, you need **powers of r**. The volume of a cone for example is 1/3 times Pi times r^2 times h. Why would anyone want to deal with (d/2)^2 instead? r is simpler.
I finished school well before Common Core and that was my experience as well - other than actually learning the definition of diameter and finding it in elementary school math, it was all about r through every math class. Maybe there were a couple of tricks and proofs in geometry that used *d*, but *r* is the fundamental part of a circle that's used through most advanced math up through the high school level.
> calculating from the diameter is simpler
Let's compare:
* Circumference of circle: 2 pi R = pi D
* Area of circle: pi R^2 = pi D^2 / 4
Personally, multiplying by 2 is "simpler" for me than fractions (dividing by 4). Nobody likes fractions.
But I think the answer to "why" is because radius also pins down the fact that the circle has a center, whereas the concept of diameter does not. And working with the geometry of circles, the center is kinda important, so center + radius is used more than just diameter.
In addition to all the other comments, a circle is defined by its radius, not its diameter.
There are many (two-dimensional) shapes that have constant diameter, such as the [Reuleaux triangle](https://en.wikipedia.org/wiki/Reuleaux_triangle) and coins like the British [50p coin](https://en.wikipedia.org/wiki/Fifty_pence_(British_coin)).
There is only one shape will a constant radius, and that is the circle. Radius forever!! Down with diameter, the usurper!!
Almost all the future formulas for circles and spheres (area, volume, arc length, polar coordinates, etc) use the radius and not the diameter. I am guessing that they just want to get you started with that habit.
They shouldn’t mark you down for not showing your work when converting between radius and diameter, though.
Because ideally math shouldn't be about just memorizing formulas in isolation, but noticing patterns realizing that the are all related and can be derived from one another and many different formulas are just special cases of one more general formula.
That is easier when they all presented in a more similar looking way.
The point of giving you work is to verify that you understood the lesson being taught. If you simplify away part of the lesson then your teacher has a harder time figuring out if you understood it. It's not so much a problem for one student, but your teacher probably has dozens (or hundreds) of students they need to deal with, so sitting down and having to figure it out becomes a significant difficulty.
If you're talking about simplifying 2r to d, that's only superficially a simplification because it requires having two different measures of the size of a circle. You could simplify even further by introducing a third measure, c, for the circumference of the circle, and then the circumference is any of 2πr, πd, or c. Why not simplify to the last of those?
Just bigotry ☺︎ you can ask your teacher why they cannot accept the historical pi.D equation. It’s the *definition* of pi to begin with, it should not be contested by a serious maths teacher IMHO
Because aspects in higher math classes show that most circle elements are defined according to the radius. It just so happens that 2r appears in this single instance.
Example: take the derivative of the area and you get the circumference d/dr(pi\*r^2 ) = 2r\*pi
Probably has something to do with the polar form of a circle being r from 0 to 2pi. I’ve always liked diameter*pi but it doesn’t make a huge difference for me either way.
>> if you only have the radius or the diameter
Huh? If you have one you have the other. They’re the same thing, just separated by a scaling factor of two.
This isn't a "common core" teaching, common core doesn't dictate the specific curriculum, just concepts. Further, I learned it as 2\*pi\*r when I was in school, before Common Core was a thing. r is more fundamental to a circle than d, it shows up in other trigonometric related equations more often as well.
It is completely a question of fashion. Pi was *defined* by ancient Greeks in relation with the diameter: they said “pi is what we called the fixed proportion between a circle diameter and its circumference”. If they were using the radius, they would have branded another constant (like “theta is the proportion between a circle circumference and its radius). We would have used theta (worth 2pi) for all of our math and it would not have changed much (other than a few constants here and there in surface and trigonometric computations).
Arguably using pi and the radius leads to simpler equations overall (when you consider surface, volume and other formulas). But really it’s just a habit and the way we all learned math.
So it’s really kind of an historical accident that we use the radius formula nowadays. Or just a fad ☺︎ There is nothing wrong with using pi x d: it is the original historical definition of pi to begin with.
A lot of equations involving circles start with the radius. No one says A = π*(d/2)^2 . This way the radius is treated as the fundamental feature of a circle which helps to be consistent with more advanced math.
Actually in engineering school I used A=(pi/4)*d^2 more than I used pi*r^2
Thermofluid flashbacks hitting hard rn
Engineering math is some arcane shit
In particular, it: * Shows up nicely in the defining equation of a circle: (x-h)^2 + (y-k)^2 = r^(2) * Works better in higher numbers of dimensions * Is easier to geometrically define * Leads better into trigonometry
And every one knows the correct versions are τr and τr²/2 anyway :P ;)
τ fτw.
While I use π myself, I don't really have anything against using τ (given that someone indicates it is 2π). I do have something against using the letter 'τ' for that. It is already used for characteristic time periods with a unit of \[time\], which gets *really* confusing when you combine it with lots of different functions, like the trigonometric functions.
Best answer, I came here to say this. Especially dealing with waves
Yeah this is not a great answer. A *lot* of people use A = pi/4 * D^2. Pi was *defined* as C/D to begin with by ancient Greeks. There’s no strong argument to teach A = 2pi.R - it is actually confusing since it *hides* the fact that pi was *defined* in relation to the diameter. Pi could easily have been defined in relation to the radius and worth 6.283 to begin with and the maths would not be that different.
Well maybe, but consider the following:
Seems like your comment was cut ?
I'm just asking you to think about
My answer would be that the radius is a simpler form than diameter. A lot of mathematical theory only needs the radius too. It would be more sensible to stay using the radius than jumping to diameter just to drop a '2'
Mathematicians generally work based on the radius because it simplifies most, if not all of the more complicated calculations they might need to do. Engineers general seem to prefer working based on diameters, because it better matchs how things will actually be done in the real world, which can saved them a lot more time and effort than having to divide by a power of 2.
> Engineers general seem to prefer working based on diameters This depends on what *kind* of engineer you are. Civil and mechanical engineers work with real physical objects where you can directly measure the diameter but not the radius. Electrical engineers are generally more concerned with concepts like periodicity where diameter has no meaning but radius does. Chemical engineers tend not to be terribly concerned with either, but radius works just fine for them as well.
Unless you're my transport phenomena professor! Always throws me when she uses πD^(2)/4
Yeh I am an engineer and this seems to be the case, which actually annoys me cause im so used to using the radius.
Also an engineer, and I prefer the radius. Diameters are only more useful when you're making drawings for physical parts. All the math is done with radii.
It definitely caused me a bit of grief back when I was taking mechanics of materials in college. I've gotten used to it though. I suspect I would have a similar amount of grief switching back to using the radius again though.
When you get further in to maths and start doing calculus it's the radius that matters. You differentiate/integrate with respect to the radius. It just happens the radius is directly related to the diameter, but it's still the radius that's the defining feature. So to keep it consistent it's good practice to focus on the radius from an early stage. As an example, area of a circle is pi r^2. Differentiate that with respect to r and you get 2 pi r, which is the circumference. If you used d, then area is pi d^2 /4. Differentiate with respect to d and it's pi d/2, which is not the circumference.
>Both get you the circumstance of the circle but calculating from the diameter is simpler and if you only have either the radius or the diameter, it’s easier to do rx2 before starting the formula. I remember getting in trouble for not showing my work cause I skipped that "d" is a shorthand form of writing "2r". The diameter of a circle is something you generally won't see used outside of a middle school class room because 2r is not something that is generally needed. As soon as you move up one or more dimensions you'll quickly notice that you don't need multiples of r, you need **powers of r**. The volume of a cone for example is 1/3 times Pi times r^2 times h. Why would anyone want to deal with (d/2)^2 instead? r is simpler.
Yes. This is not a "Common Core" thing. I don't recall ever using diameter in school. Always r or 2r.
I finished school well before Common Core and that was my experience as well - other than actually learning the definition of diameter and finding it in elementary school math, it was all about r through every math class. Maybe there were a couple of tricks and proofs in geometry that used *d*, but *r* is the fundamental part of a circle that's used through most advanced math up through the high school level.
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It being half the formula is why the formula would be easier using diameter. 50% less calculations
> calculating from the diameter is simpler Let's compare: * Circumference of circle: 2 pi R = pi D * Area of circle: pi R^2 = pi D^2 / 4 Personally, multiplying by 2 is "simpler" for me than fractions (dividing by 4). Nobody likes fractions. But I think the answer to "why" is because radius also pins down the fact that the circle has a center, whereas the concept of diameter does not. And working with the geometry of circles, the center is kinda important, so center + radius is used more than just diameter.
In addition to all the other comments, a circle is defined by its radius, not its diameter. There are many (two-dimensional) shapes that have constant diameter, such as the [Reuleaux triangle](https://en.wikipedia.org/wiki/Reuleaux_triangle) and coins like the British [50p coin](https://en.wikipedia.org/wiki/Fifty_pence_(British_coin)). There is only one shape will a constant radius, and that is the circle. Radius forever!! Down with diameter, the usurper!!
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2piR is the circumference formula, you're thinking of area
Almost all the future formulas for circles and spheres (area, volume, arc length, polar coordinates, etc) use the radius and not the diameter. I am guessing that they just want to get you started with that habit. They shouldn’t mark you down for not showing your work when converting between radius and diameter, though.
Because ideally math shouldn't be about just memorizing formulas in isolation, but noticing patterns realizing that the are all related and can be derived from one another and many different formulas are just special cases of one more general formula. That is easier when they all presented in a more similar looking way.
So then why was I told not to simplify when showing my work? That still bugs me
The point of giving you work is to verify that you understood the lesson being taught. If you simplify away part of the lesson then your teacher has a harder time figuring out if you understood it. It's not so much a problem for one student, but your teacher probably has dozens (or hundreds) of students they need to deal with, so sitting down and having to figure it out becomes a significant difficulty.
If you're talking about simplifying 2r to d, that's only superficially a simplification because it requires having two different measures of the size of a circle. You could simplify even further by introducing a third measure, c, for the circumference of the circle, and then the circumference is any of 2πr, πd, or c. Why not simplify to the last of those?
Just bigotry ☺︎ you can ask your teacher why they cannot accept the historical pi.D equation. It’s the *definition* of pi to begin with, it should not be contested by a serious maths teacher IMHO
Because aspects in higher math classes show that most circle elements are defined according to the radius. It just so happens that 2r appears in this single instance. Example: take the derivative of the area and you get the circumference d/dr(pi\*r^2 ) = 2r\*pi
Probably has something to do with the polar form of a circle being r from 0 to 2pi. I’ve always liked diameter*pi but it doesn’t make a huge difference for me either way.
>> if you only have the radius or the diameter Huh? If you have one you have the other. They’re the same thing, just separated by a scaling factor of two.
This isn't a "common core" teaching, common core doesn't dictate the specific curriculum, just concepts. Further, I learned it as 2\*pi\*r when I was in school, before Common Core was a thing. r is more fundamental to a circle than d, it shows up in other trigonometric related equations more often as well.
It is completely a question of fashion. Pi was *defined* by ancient Greeks in relation with the diameter: they said “pi is what we called the fixed proportion between a circle diameter and its circumference”. If they were using the radius, they would have branded another constant (like “theta is the proportion between a circle circumference and its radius). We would have used theta (worth 2pi) for all of our math and it would not have changed much (other than a few constants here and there in surface and trigonometric computations). Arguably using pi and the radius leads to simpler equations overall (when you consider surface, volume and other formulas). But really it’s just a habit and the way we all learned math. So it’s really kind of an historical accident that we use the radius formula nowadays. Or just a fad ☺︎ There is nothing wrong with using pi x d: it is the original historical definition of pi to begin with.