Question based on perimeter of a ring
$begingroup$
Why the perimeter of a ring is $2pi(R+r)$ and not $2pi(R-r)$ ? where $R$ is the bigger circle radius and $r$ is smaller circle radius
geometry gmat-exam
asked Aug 2 '17 at 18:32
EmilySekuzEmilySekuz
5728
$endgroup$
add a comment |
$begingroup$
Why the perimeter of a ring is $2pi(R+r)$ and not $2pi(R-r)$ ? where $R$ is the bigger circle radius and $r$ is smaller circle radius
geometry gmat-exam
asked Aug 2 '17 at 18:32
EmilySekuzEmilySekuz
5728
$endgroup$
add a comment |
$begingroup$
Why the perimeter of a ring is $2pi(R+r)$ and not $2pi(R-r)$ ? where $R$ is the bigger circle radius and $r$ is smaller circle radius
geometry gmat-exam
asked Aug 2 '17 at 18:32
EmilySekuzEmilySekuz
5728
$endgroup$
Why the perimeter of a ring is $2pi(R+r)$ and not $2pi(R-r)$ ? where $R$ is the bigger circle radius and $r$ is smaller circle radius
geometry gmat-exam
geometry gmat-exam
asked Aug 2 '17 at 18:32
EmilySekuzEmilySekuz
5728
asked Aug 2 '17 at 18:32
EmilySekuzEmilySekuz
5728
asked Aug 2 '17 at 18:32
EmilySekuzEmilySekuz
5728
asked Aug 2 '17 at 18:32
EmilySekuzEmilySekuz
5728
asked Aug 2 '17 at 18:32
EmilySekuzEmilySekuz
5728
5728
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
It is the sum of the inner perimeter and the outer perimeter, not the difference between the two.
answered Aug 2 '17 at 18:33
Eric TowersEric Towers
33.4k22370
$endgroup$
add a comment |
$begingroup$
The basic reason is that the outer radius of the ring is $R+r$. In order to make this easier to discuss, pretend that we are talking about a metal ring that you are wearing on your finger. To get to the outer edge of the metal, you have to first travel from the center of your finger to the middle of the metal part ($R$) and then from the middle of the metal part to the outer edge of the metal part ($r$). So the outer edge of the ring is a circle with radius $R+r$, so its circumference is $2pi(R+r)$.
I have added this diagram for clarity. As noted in the comments below, I think that you and the author of the result that you are talking about have different ideas about what the "bigger circle" and "smaller circle" are, and possibly about what the "perimeter" means in this case. This is my interpretation. I have marked the perimeter as $p$, as well as marking $r$ and $R$.
answered Aug 2 '17 at 18:42
sasquiressasquires
357211
$endgroup$
$begingroup$
As given, "$R$ is the bigger circle radius", so the outer radius of the ring is $R$. Your calculation also only finds the outer circle's contribution to the perimeter. If the given ring had been as you describe, with outer radius $R + r$ and inner radius $R - r$, you have not included the $2 pi (R - r)$ perimeter of the inner circle.
$endgroup$
– Eric Towers
Aug 7 '17 at 4:59
$begingroup$
I think that the whole problem is that we are talking about different perimeters, and we probably also have different definitions of "bigger circle radius" and "smaller circle radius". I am editing my answer with a picture of the definitions that I (and probably whoever wrote the original $2pi (R+r)$ result that inspired your question) am using. If you are using a different definition, then of course you will get a different answer.
$endgroup$
– sasquires
Aug 8 '17 at 18:26
1
$begingroup$
Perimeter: the boundary of a closed geometric figure. The figure in question has two boundary components. You have only provided the length of the outer of the two components. If the requested quantity had been the circumference, perhaps ignoring the inner component would be defensible. Nevertheless, the perimeter of the annulus you have drawn is $2pi(R+r) + 2pi(R-r) = 4 pi R$.
$endgroup$
– Eric Towers
Aug 9 '17 at 13:14
$begingroup$
@EricTowers The question is extremely vague to begin with. For example, it is not even clear whether "ring" means a torus or an annulus. For an annulus, you are correct, and for a torus, then the perimeter is ill-defined. But I suspect the the OP was given this formula and was confused by it, and I think that my diagram probably clarified things for her. (Incidentally: sorry, I previously mistook you for the OP.) I would not at all be surprised if a high school teacher conflated perimeter and outer circumference, and the OP was asking for a justification of the formula she was given.
$endgroup$
– sasquires
Aug 11 '17 at 17:27
add a comment |
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2 Answers
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2 Answers
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$begingroup$
It is the sum of the inner perimeter and the outer perimeter, not the difference between the two.
answered Aug 2 '17 at 18:33
Eric TowersEric Towers
33.4k22370
$endgroup$
add a comment |
$begingroup$
It is the sum of the inner perimeter and the outer perimeter, not the difference between the two.
answered Aug 2 '17 at 18:33
Eric TowersEric Towers
33.4k22370
$endgroup$
add a comment |
$begingroup$
It is the sum of the inner perimeter and the outer perimeter, not the difference between the two.
answered Aug 2 '17 at 18:33
Eric TowersEric Towers
33.4k22370
$endgroup$
It is the sum of the inner perimeter and the outer perimeter, not the difference between the two.
answered Aug 2 '17 at 18:33
Eric TowersEric Towers
33.4k22370
answered Aug 2 '17 at 18:33
Eric TowersEric Towers
33.4k22370
answered Aug 2 '17 at 18:33
Eric TowersEric Towers
33.4k22370
answered Aug 2 '17 at 18:33
Eric TowersEric Towers
33.4k22370
33.4k22370
add a comment |
add a comment |
$begingroup$
The basic reason is that the outer radius of the ring is $R+r$. In order to make this easier to discuss, pretend that we are talking about a metal ring that you are wearing on your finger. To get to the outer edge of the metal, you have to first travel from the center of your finger to the middle of the metal part ($R$) and then from the middle of the metal part to the outer edge of the metal part ($r$). So the outer edge of the ring is a circle with radius $R+r$, so its circumference is $2pi(R+r)$.
I have added this diagram for clarity. As noted in the comments below, I think that you and the author of the result that you are talking about have different ideas about what the "bigger circle" and "smaller circle" are, and possibly about what the "perimeter" means in this case. This is my interpretation. I have marked the perimeter as $p$, as well as marking $r$ and $R$.
answered Aug 2 '17 at 18:42
sasquiressasquires
357211
$endgroup$
$begingroup$
As given, "$R$ is the bigger circle radius", so the outer radius of the ring is $R$. Your calculation also only finds the outer circle's contribution to the perimeter. If the given ring had been as you describe, with outer radius $R + r$ and inner radius $R - r$, you have not included the $2 pi (R - r)$ perimeter of the inner circle.
$endgroup$
– Eric Towers
Aug 7 '17 at 4:59
$begingroup$
I think that the whole problem is that we are talking about different perimeters, and we probably also have different definitions of "bigger circle radius" and "smaller circle radius". I am editing my answer with a picture of the definitions that I (and probably whoever wrote the original $2pi (R+r)$ result that inspired your question) am using. If you are using a different definition, then of course you will get a different answer.
$endgroup$
– sasquires
Aug 8 '17 at 18:26
1
$begingroup$
Perimeter: the boundary of a closed geometric figure. The figure in question has two boundary components. You have only provided the length of the outer of the two components. If the requested quantity had been the circumference, perhaps ignoring the inner component would be defensible. Nevertheless, the perimeter of the annulus you have drawn is $2pi(R+r) + 2pi(R-r) = 4 pi R$.
$endgroup$
– Eric Towers
Aug 9 '17 at 13:14
$begingroup$
@EricTowers The question is extremely vague to begin with. For example, it is not even clear whether "ring" means a torus or an annulus. For an annulus, you are correct, and for a torus, then the perimeter is ill-defined. But I suspect the the OP was given this formula and was confused by it, and I think that my diagram probably clarified things for her. (Incidentally: sorry, I previously mistook you for the OP.) I would not at all be surprised if a high school teacher conflated perimeter and outer circumference, and the OP was asking for a justification of the formula she was given.
$endgroup$
– sasquires
Aug 11 '17 at 17:27
add a comment |
$begingroup$
The basic reason is that the outer radius of the ring is $R+r$. In order to make this easier to discuss, pretend that we are talking about a metal ring that you are wearing on your finger. To get to the outer edge of the metal, you have to first travel from the center of your finger to the middle of the metal part ($R$) and then from the middle of the metal part to the outer edge of the metal part ($r$). So the outer edge of the ring is a circle with radius $R+r$, so its circumference is $2pi(R+r)$.
I have added this diagram for clarity. As noted in the comments below, I think that you and the author of the result that you are talking about have different ideas about what the "bigger circle" and "smaller circle" are, and possibly about what the "perimeter" means in this case. This is my interpretation. I have marked the perimeter as $p$, as well as marking $r$ and $R$.
answered Aug 2 '17 at 18:42
sasquiressasquires
357211
$endgroup$
$begingroup$
As given, "$R$ is the bigger circle radius", so the outer radius of the ring is $R$. Your calculation also only finds the outer circle's contribution to the perimeter. If the given ring had been as you describe, with outer radius $R + r$ and inner radius $R - r$, you have not included the $2 pi (R - r)$ perimeter of the inner circle.
$endgroup$
– Eric Towers
Aug 7 '17 at 4:59
$begingroup$
I think that the whole problem is that we are talking about different perimeters, and we probably also have different definitions of "bigger circle radius" and "smaller circle radius". I am editing my answer with a picture of the definitions that I (and probably whoever wrote the original $2pi (R+r)$ result that inspired your question) am using. If you are using a different definition, then of course you will get a different answer.
$endgroup$
– sasquires
Aug 8 '17 at 18:26
1
$begingroup$
Perimeter: the boundary of a closed geometric figure. The figure in question has two boundary components. You have only provided the length of the outer of the two components. If the requested quantity had been the circumference, perhaps ignoring the inner component would be defensible. Nevertheless, the perimeter of the annulus you have drawn is $2pi(R+r) + 2pi(R-r) = 4 pi R$.
$endgroup$
– Eric Towers
Aug 9 '17 at 13:14
$begingroup$
@EricTowers The question is extremely vague to begin with. For example, it is not even clear whether "ring" means a torus or an annulus. For an annulus, you are correct, and for a torus, then the perimeter is ill-defined. But I suspect the the OP was given this formula and was confused by it, and I think that my diagram probably clarified things for her. (Incidentally: sorry, I previously mistook you for the OP.) I would not at all be surprised if a high school teacher conflated perimeter and outer circumference, and the OP was asking for a justification of the formula she was given.
$endgroup$
– sasquires
Aug 11 '17 at 17:27
add a comment |
$begingroup$
The basic reason is that the outer radius of the ring is $R+r$. In order to make this easier to discuss, pretend that we are talking about a metal ring that you are wearing on your finger. To get to the outer edge of the metal, you have to first travel from the center of your finger to the middle of the metal part ($R$) and then from the middle of the metal part to the outer edge of the metal part ($r$). So the outer edge of the ring is a circle with radius $R+r$, so its circumference is $2pi(R+r)$.
I have added this diagram for clarity. As noted in the comments below, I think that you and the author of the result that you are talking about have different ideas about what the "bigger circle" and "smaller circle" are, and possibly about what the "perimeter" means in this case. This is my interpretation. I have marked the perimeter as $p$, as well as marking $r$ and $R$.
answered Aug 2 '17 at 18:42
sasquiressasquires
357211
$endgroup$
The basic reason is that the outer radius of the ring is $R+r$. In order to make this easier to discuss, pretend that we are talking about a metal ring that you are wearing on your finger. To get to the outer edge of the metal, you have to first travel from the center of your finger to the middle of the metal part ($R$) and then from the middle of the metal part to the outer edge of the metal part ($r$). So the outer edge of the ring is a circle with radius $R+r$, so its circumference is $2pi(R+r)$.
I have added this diagram for clarity. As noted in the comments below, I think that you and the author of the result that you are talking about have different ideas about what the "bigger circle" and "smaller circle" are, and possibly about what the "perimeter" means in this case. This is my interpretation. I have marked the perimeter as $p$, as well as marking $r$ and $R$.
answered Aug 2 '17 at 18:42
sasquiressasquires
357211
edited Aug 8 '17 at 18:35
answered Aug 2 '17 at 18:42
sasquiressasquires
357211
answered Aug 2 '17 at 18:42
sasquiressasquires
357211
answered Aug 2 '17 at 18:42
sasquiressasquires
357211
357211
$begingroup$
As given, "$R$ is the bigger circle radius", so the outer radius of the ring is $R$. Your calculation also only finds the outer circle's contribution to the perimeter. If the given ring had been as you describe, with outer radius $R + r$ and inner radius $R - r$, you have not included the $2 pi (R - r)$ perimeter of the inner circle.
$endgroup$
– Eric Towers
Aug 7 '17 at 4:59
$begingroup$
I think that the whole problem is that we are talking about different perimeters, and we probably also have different definitions of "bigger circle radius" and "smaller circle radius". I am editing my answer with a picture of the definitions that I (and probably whoever wrote the original $2pi (R+r)$ result that inspired your question) am using. If you are using a different definition, then of course you will get a different answer.
$endgroup$
– sasquires
Aug 8 '17 at 18:26
1
$begingroup$
Perimeter: the boundary of a closed geometric figure. The figure in question has two boundary components. You have only provided the length of the outer of the two components. If the requested quantity had been the circumference, perhaps ignoring the inner component would be defensible. Nevertheless, the perimeter of the annulus you have drawn is $2pi(R+r) + 2pi(R-r) = 4 pi R$.
$endgroup$
– Eric Towers
Aug 9 '17 at 13:14
$begingroup$
@EricTowers The question is extremely vague to begin with. For example, it is not even clear whether "ring" means a torus or an annulus. For an annulus, you are correct, and for a torus, then the perimeter is ill-defined. But I suspect the the OP was given this formula and was confused by it, and I think that my diagram probably clarified things for her. (Incidentally: sorry, I previously mistook you for the OP.) I would not at all be surprised if a high school teacher conflated perimeter and outer circumference, and the OP was asking for a justification of the formula she was given.
$endgroup$
– sasquires
Aug 11 '17 at 17:27
add a comment |
$begingroup$
As given, "$R$ is the bigger circle radius", so the outer radius of the ring is $R$. Your calculation also only finds the outer circle's contribution to the perimeter. If the given ring had been as you describe, with outer radius $R + r$ and inner radius $R - r$, you have not included the $2 pi (R - r)$ perimeter of the inner circle.
$endgroup$
– Eric Towers
Aug 7 '17 at 4:59
$begingroup$
I think that the whole problem is that we are talking about different perimeters, and we probably also have different definitions of "bigger circle radius" and "smaller circle radius". I am editing my answer with a picture of the definitions that I (and probably whoever wrote the original $2pi (R+r)$ result that inspired your question) am using. If you are using a different definition, then of course you will get a different answer.
$endgroup$
– sasquires
Aug 8 '17 at 18:26
1
$begingroup$
Perimeter: the boundary of a closed geometric figure. The figure in question has two boundary components. You have only provided the length of the outer of the two components. If the requested quantity had been the circumference, perhaps ignoring the inner component would be defensible. Nevertheless, the perimeter of the annulus you have drawn is $2pi(R+r) + 2pi(R-r) = 4 pi R$.
$endgroup$
– Eric Towers
Aug 9 '17 at 13:14
$begingroup$
@EricTowers The question is extremely vague to begin with. For example, it is not even clear whether "ring" means a torus or an annulus. For an annulus, you are correct, and for a torus, then the perimeter is ill-defined. But I suspect the the OP was given this formula and was confused by it, and I think that my diagram probably clarified things for her. (Incidentally: sorry, I previously mistook you for the OP.) I would not at all be surprised if a high school teacher conflated perimeter and outer circumference, and the OP was asking for a justification of the formula she was given.
$endgroup$
– sasquires
Aug 11 '17 at 17:27
$begingroup$
As given, "$R$ is the bigger circle radius", so the outer radius of the ring is $R$. Your calculation also only finds the outer circle's contribution to the perimeter. If the given ring had been as you describe, with outer radius $R + r$ and inner radius $R - r$, you have not included the $2 pi (R - r)$ perimeter of the inner circle.
$endgroup$
– Eric Towers
Aug 7 '17 at 4:59
$begingroup$
As given, "$R$ is the bigger circle radius", so the outer radius of the ring is $R$. Your calculation also only finds the outer circle's contribution to the perimeter. If the given ring had been as you describe, with outer radius $R + r$ and inner radius $R - r$, you have not included the $2 pi (R - r)$ perimeter of the inner circle.
$endgroup$
– Eric Towers
Aug 7 '17 at 4:59
$begingroup$
I think that the whole problem is that we are talking about different perimeters, and we probably also have different definitions of "bigger circle radius" and "smaller circle radius". I am editing my answer with a picture of the definitions that I (and probably whoever wrote the original $2pi (R+r)$ result that inspired your question) am using. If you are using a different definition, then of course you will get a different answer.
$endgroup$
– sasquires
Aug 8 '17 at 18:26
$begingroup$
I think that the whole problem is that we are talking about different perimeters, and we probably also have different definitions of "bigger circle radius" and "smaller circle radius". I am editing my answer with a picture of the definitions that I (and probably whoever wrote the original $2pi (R+r)$ result that inspired your question) am using. If you are using a different definition, then of course you will get a different answer.
$endgroup$
– sasquires
Aug 8 '17 at 18:26
1
1
$begingroup$
Perimeter: the boundary of a closed geometric figure. The figure in question has two boundary components. You have only provided the length of the outer of the two components. If the requested quantity had been the circumference, perhaps ignoring the inner component would be defensible. Nevertheless, the perimeter of the annulus you have drawn is $2pi(R+r) + 2pi(R-r) = 4 pi R$.
$endgroup$
– Eric Towers
Aug 9 '17 at 13:14
$begingroup$
Perimeter: the boundary of a closed geometric figure. The figure in question has two boundary components. You have only provided the length of the outer of the two components. If the requested quantity had been the circumference, perhaps ignoring the inner component would be defensible. Nevertheless, the perimeter of the annulus you have drawn is $2pi(R+r) + 2pi(R-r) = 4 pi R$.
$endgroup$
– Eric Towers
Aug 9 '17 at 13:14
$begingroup$
@EricTowers The question is extremely vague to begin with. For example, it is not even clear whether "ring" means a torus or an annulus. For an annulus, you are correct, and for a torus, then the perimeter is ill-defined. But I suspect the the OP was given this formula and was confused by it, and I think that my diagram probably clarified things for her. (Incidentally: sorry, I previously mistook you for the OP.) I would not at all be surprised if a high school teacher conflated perimeter and outer circumference, and the OP was asking for a justification of the formula she was given.
$endgroup$
– sasquires
Aug 11 '17 at 17:27
$begingroup$
@EricTowers The question is extremely vague to begin with. For example, it is not even clear whether "ring" means a torus or an annulus. For an annulus, you are correct, and for a torus, then the perimeter is ill-defined. But I suspect the the OP was given this formula and was confused by it, and I think that my diagram probably clarified things for her. (Incidentally: sorry, I previously mistook you for the OP.) I would not at all be surprised if a high school teacher conflated perimeter and outer circumference, and the OP was asking for a justification of the formula she was given.
$endgroup$
– sasquires
Aug 11 '17 at 17:27
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