A flower in a hexagon
$begingroup$
The area of the flower
So I had a geometry problem in my maths exam test and I couldn't solve it, though it didn't seem that hard. I tried to connect the points from center to end and create an equilateral triangle but still was unable to find the solution.
Would be thankful if you could help me out.
The question is the following:
We have an equal hexagon with sides equal to $1$ and six circular arcs with radius equal to $1$ from each vertices of the regular hexagon in it create a flower-shaped-like object. Find the area of the flower.
Here are the possible answers ↓
1. $pi$
3. $3pi/2$
4. $4sqrt2-pi$
5. $pi/2+sqrt3$
6. $2pi-3sqrt3$
geometry logic triangle area
$endgroup$
add a comment |
$begingroup$
The area of the flower
So I had a geometry problem in my maths exam test and I couldn't solve it, though it didn't seem that hard. I tried to connect the points from center to end and create an equilateral triangle but still was unable to find the solution.
Would be thankful if you could help me out.
The question is the following:
We have an equal hexagon with sides equal to $1$ and six circular arcs with radius equal to $1$ from each vertices of the regular hexagon in it create a flower-shaped-like object. Find the area of the flower.
Here are the possible answers ↓
1. $pi$
3. $3pi/2$
4. $4sqrt2-pi$
5. $pi/2+sqrt3$
6. $2pi-3sqrt3$
geometry logic triangle area
$endgroup$
2
$begingroup$
I count six ellipses
$endgroup$
– Ross Millikan
5 hours ago
$begingroup$
Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
$endgroup$
– Oscar Lanzi
4 hours ago
add a comment |
$begingroup$
The area of the flower
So I had a geometry problem in my maths exam test and I couldn't solve it, though it didn't seem that hard. I tried to connect the points from center to end and create an equilateral triangle but still was unable to find the solution.
Would be thankful if you could help me out.
The question is the following:
We have an equal hexagon with sides equal to $1$ and six circular arcs with radius equal to $1$ from each vertices of the regular hexagon in it create a flower-shaped-like object. Find the area of the flower.
Here are the possible answers ↓
1. $pi$
3. $3pi/2$
4. $4sqrt2-pi$
5. $pi/2+sqrt3$
6. $2pi-3sqrt3$
geometry logic triangle area
$endgroup$
The area of the flower
So I had a geometry problem in my maths exam test and I couldn't solve it, though it didn't seem that hard. I tried to connect the points from center to end and create an equilateral triangle but still was unable to find the solution.
Would be thankful if you could help me out.
The question is the following:
We have an equal hexagon with sides equal to $1$ and six circular arcs with radius equal to $1$ from each vertices of the regular hexagon in it create a flower-shaped-like object. Find the area of the flower.
Here are the possible answers ↓
1. $pi$
3. $3pi/2$
4. $4sqrt2-pi$
5. $pi/2+sqrt3$
6. $2pi-3sqrt3$
geometry logic triangle area
geometry logic triangle area
edited 1 hour ago
J. W. Tanner
2,6161217
2,6161217
asked 6 hours ago
DAVODAVO
135
135
2
$begingroup$
I count six ellipses
$endgroup$
– Ross Millikan
5 hours ago
$begingroup$
Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
$endgroup$
– Oscar Lanzi
4 hours ago
add a comment |
2
$begingroup$
I count six ellipses
$endgroup$
– Ross Millikan
5 hours ago
$begingroup$
Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
$endgroup$
– Oscar Lanzi
4 hours ago
2
2
$begingroup$
I count six ellipses
$endgroup$
– Ross Millikan
5 hours ago
$begingroup$
I count six ellipses
$endgroup$
– Ross Millikan
5 hours ago
$begingroup$
Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
$endgroup$
– Oscar Lanzi
4 hours ago
$begingroup$
Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
$endgroup$
– Oscar Lanzi
4 hours ago
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.
The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.
The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.
There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.
$endgroup$
$begingroup$
Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
$endgroup$
– Trebor
4 hours ago
$begingroup$
@Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
$endgroup$
– Deepak
4 hours ago
$begingroup$
@AnirbanNiloy Thank you for the nice diagram. :)
$endgroup$
– Deepak
12 mins ago
$begingroup$
@Deepak Welcome! Anytime!!!! You deserve more than that.
$endgroup$
– Anirban Niloy
10 mins ago
add a comment |
$begingroup$
We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.
If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.
Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.
And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.
$endgroup$
add a comment |
$begingroup$
Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.
What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:
$3p + 2w = {frac 1 3} pi r^2 = frac pi 3$
What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:
$p + w = frac{sqrt 3}{4}$
Solving the two simultaneous equations:
$2p + 2w = frac{sqrt 3}{2}$
$p = frac{pi}{3} - frac{sqrt 3}{2}$
And the area of the flower is $6p = 2pi - 3sqrt 3$.
New contributor
$endgroup$
add a comment |
Your Answer
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3 Answers
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active
oldest
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3 Answers
3
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.
The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.
The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.
There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.
$endgroup$
$begingroup$
Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
$endgroup$
– Trebor
4 hours ago
$begingroup$
@Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
$endgroup$
– Deepak
4 hours ago
$begingroup$
@AnirbanNiloy Thank you for the nice diagram. :)
$endgroup$
– Deepak
12 mins ago
$begingroup$
@Deepak Welcome! Anytime!!!! You deserve more than that.
$endgroup$
– Anirban Niloy
10 mins ago
add a comment |
$begingroup$
Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.
The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.
The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.
There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.
$endgroup$
$begingroup$
Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
$endgroup$
– Trebor
4 hours ago
$begingroup$
@Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
$endgroup$
– Deepak
4 hours ago
$begingroup$
@AnirbanNiloy Thank you for the nice diagram. :)
$endgroup$
– Deepak
12 mins ago
$begingroup$
@Deepak Welcome! Anytime!!!! You deserve more than that.
$endgroup$
– Anirban Niloy
10 mins ago
add a comment |
$begingroup$
Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.
The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.
The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.
There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.
$endgroup$
Assuming those are circular arcs and not "ellipses", you can indeed find the area exactly.
The "flower" has six "petals" Each of those petals has axial symmetry and can be divided into two halves. Each of those halves is the segment subtending a central angle of $frac{pi}{3}$ (radian measure) of a circle of radius $1$.
The area of one such segment is $frac 12 r^2(theta - sintheta) = frac 12(frac{pi}{3} - frac{sqrt{3}}{2})$.
There are $12$ such segments, yielding the total area of the "flower" as $2pi - 3sqrt 3$.
edited 2 hours ago
Anirban Niloy
609218
609218
answered 4 hours ago
DeepakDeepak
17.2k11536
17.2k11536
$begingroup$
Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
$endgroup$
– Trebor
4 hours ago
$begingroup$
@Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
$endgroup$
– Deepak
4 hours ago
$begingroup$
@AnirbanNiloy Thank you for the nice diagram. :)
$endgroup$
– Deepak
12 mins ago
$begingroup$
@Deepak Welcome! Anytime!!!! You deserve more than that.
$endgroup$
– Anirban Niloy
10 mins ago
add a comment |
$begingroup$
Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
$endgroup$
– Trebor
4 hours ago
$begingroup$
@Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
$endgroup$
– Deepak
4 hours ago
$begingroup$
@AnirbanNiloy Thank you for the nice diagram. :)
$endgroup$
– Deepak
12 mins ago
$begingroup$
@Deepak Welcome! Anytime!!!! You deserve more than that.
$endgroup$
– Anirban Niloy
10 mins ago
$begingroup$
Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
$endgroup$
– Trebor
4 hours ago
$begingroup$
Since the shape of each petal is claimed to be ellipses, your answer is not valid in assuming that they consist of circular arcs. However, I do reckon that they are intented to be arcs.
$endgroup$
– Trebor
4 hours ago
$begingroup$
@Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
$endgroup$
– Deepak
4 hours ago
$begingroup$
@Trebor I had edited my answer (I think before your comment) to clarify that assumption. However, I think it's a valid assumption, and just a very poor question.
$endgroup$
– Deepak
4 hours ago
$begingroup$
@AnirbanNiloy Thank you for the nice diagram. :)
$endgroup$
– Deepak
12 mins ago
$begingroup$
@AnirbanNiloy Thank you for the nice diagram. :)
$endgroup$
– Deepak
12 mins ago
$begingroup$
@Deepak Welcome! Anytime!!!! You deserve more than that.
$endgroup$
– Anirban Niloy
10 mins ago
$begingroup$
@Deepak Welcome! Anytime!!!! You deserve more than that.
$endgroup$
– Anirban Niloy
10 mins ago
add a comment |
$begingroup$
We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.
If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.
Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.
And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.
$endgroup$
add a comment |
$begingroup$
We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.
If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.
Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.
And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.
$endgroup$
add a comment |
$begingroup$
We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.
If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.
Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.
And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.
$endgroup$
We can't easily find the area exactly - we just don't have enough information for that. But then, what are the estimates we're looking at? (1) $approx 3.1$. (3) $approx 4.7$. (4) $approx 2.5$. (5) $approx 3.3$. (6) $approx 1.1$.
If the area's close to $3$, we might have difficulty deciding. If it's farther away, it'll be an easy choice.
Now, what's the area of the hexagon? That's something we can calculate exactly; an equilateral triangle of side $1$ has area $frac{sqrt{3}}{4}$, and the hexagon is six of those stuck together for a total area of $frac32sqrt{3}approx 2.6$. The flower is clearly much less than that. Only one of the answers is plausible. It has to be (6), $2pi-3sqrt{3}$.
And, reverse-engineering from the answer... the curves that define the "petals" are the six circles centered at the vertices of the hexagon with radius $1$. That bit about "five ellipses" is just wrong.
answered 5 hours ago
jmerryjmerry
11.6k1527
11.6k1527
add a comment |
add a comment |
$begingroup$
Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.
What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:
$3p + 2w = {frac 1 3} pi r^2 = frac pi 3$
What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:
$p + w = frac{sqrt 3}{4}$
Solving the two simultaneous equations:
$2p + 2w = frac{sqrt 3}{2}$
$p = frac{pi}{3} - frac{sqrt 3}{2}$
And the area of the flower is $6p = 2pi - 3sqrt 3$.
New contributor
$endgroup$
add a comment |
$begingroup$
Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.
What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:
$3p + 2w = {frac 1 3} pi r^2 = frac pi 3$
What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:
$p + w = frac{sqrt 3}{4}$
Solving the two simultaneous equations:
$2p + 2w = frac{sqrt 3}{2}$
$p = frac{pi}{3} - frac{sqrt 3}{2}$
And the area of the flower is $6p = 2pi - 3sqrt 3$.
New contributor
$endgroup$
add a comment |
$begingroup$
Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.
What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:
$3p + 2w = {frac 1 3} pi r^2 = frac pi 3$
What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:
$p + w = frac{sqrt 3}{4}$
Solving the two simultaneous equations:
$2p + 2w = frac{sqrt 3}{2}$
$p = frac{pi}{3} - frac{sqrt 3}{2}$
And the area of the flower is $6p = 2pi - 3sqrt 3$.
New contributor
$endgroup$
Here's a solution that doesn't require trigonometry. The hexagon is made up of 6 "petals" and 6 "wedges" (the scooped out triangles between the petals). Call the area of a petal $p$, and the area of a wedge $w$.
What is the area enclosed by two adjacent sides of the hexagon and the arc drawn from the vertex between them. (For example, the region EFAGE). It's one third of a unit circle, and contains three petals and two wedges:
$3p + 2w = {frac 1 3} pi r^2 = frac pi 3$
What is the area of one triangle (one sixth of the hexagon). The altitude of the triangle is $sqrt{1 - {frac 1 4}} = frac{sqrt 3}{2}$, so the area is $frac{sqrt 3}{4}$. The triangle is made up of two half-petals and one wedge:
$p + w = frac{sqrt 3}{4}$
Solving the two simultaneous equations:
$2p + 2w = frac{sqrt 3}{2}$
$p = frac{pi}{3} - frac{sqrt 3}{2}$
And the area of the flower is $6p = 2pi - 3sqrt 3$.
New contributor
New contributor
answered 25 mins ago
AndyBAndyB
1
1
New contributor
New contributor
add a comment |
add a comment |
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2
$begingroup$
I count six ellipses
$endgroup$
– Ross Millikan
5 hours ago
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Are the ellipses tangent to the sides at the vertices and meeting without overlap at the center? If so, they are not ellipses. Do you mean pairs of circular arcs?
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– Oscar Lanzi
4 hours ago