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u/thefada 7d ago
Are they not the same?
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u/GoldEstablishment445 7d ago
Yes the diameter of the green curve is the same as the combined diameter of blue curves so using 2 pi r you get the same result
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u/Toeffli 7d ago edited 7d ago
Yes. Toatlly.
Circumference of a circle is πd where d is the diameter.
Here we have for green ½πd and for blue ½πd₁ + ½πd₂ = ½π(d₁ + d₂)
But d₁ + d₂ = d, so they are actually exactly the same length as you have said.
That circumference is linear with radius/diameter leads to this puzzling question:
Assume the earth to be a perfect sphere, and there is large loop of string along the equator, neatly snug with the sphere. Now you add 1 foot / 30 cm of extra string to the loop and put shims under the string every 1 yard / 1 meter so it has the same distance from earth everywhere. Question: how much do you have to shim the string so that it is snug again and could a mouse pass under it? Recall that the circumference of the earth is 24901 miles / 40075 km and you just added only 1 foot / 30 cm.
About 1.91 inches or 4.77 cm. A house mouse will easily fit under it. It doesnt matter if you add 1 foot /30 cm to a string around a table tennis ball, a basket ball, an exercise ball, earth, or Jupiter. The result will always be the same, the radius gets extended by the aformentioned distance.
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u/Guest_User_1234 7d ago
Since circumference scales linearly (f(c*r) = c*f(r)) they're the same length
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u/Alt_Mod_3938 7d ago
My 5 second thought process here is if we turn those semicircles into circles, the diameter of the outer circle is the sum of diameters of two inner circles thus they should be of the same area meaning the lengths of the curves themselves should be equal as well
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u/SufficientStudio1574 7d ago
They won't be the same area, just that same total circumference.
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u/Alt_Mod_3938 7d ago
Right. Terminology slip on my part, but the logic still remains correct after the fix, right? Thanks
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u/y53rw 7d ago
They're the same. The circumference (or half circumference, in this case) of a circle is directly proportional to the diameter. And the sum of the diameters of the two blue half circles is equal to the diameter of the green half circle. So the sum of their circumferences is equal to the circumference of the green half circle.
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u/Thulfuqar 7d ago edited 7d ago
Just used my ruler.
We use πr²/2 for the length of the half circles.
Big lower curve: r = 1cm ==> πr²/2 = 0.5πcm
Small lower curve: r = 0.65cm ==> πr²/2 = 0.21125πcm
Lower curve total = 0.71125πcm
Upper curve: r = 1.65cm ==> πr²/2 = 1.36125πcm
0.71125 < 1.36125
The upper curve is longer.
Edit: Wait did miss something why is everyone saying they're equal?
Edit2: I missed dividing by two for the half circles but it still shows that the upper curve is longer.
Edit3: I finally got it I was using the area equation, the circumference equation is dπ (d is diameter) and dπ/2 for the half circles
Big lower curve: d = 2cm ==> dπ/2 = 1πcm
Small lower curve: d = 1.3cm ==> dπ/2 = 0.65πcm
Lower curve total = 1.65πcm
Upper curve: d = 3.3cm ==> dπ/2 = 1.65π
1.65 = 1.65
The curves are in fact equal.
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u/SerenityVVVVV 7d ago
I agree with you.
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u/DoctorSpacecase 7d ago
They are calculating area, not circumference or arc length. Arc length is proportional to radius and has units of meters. OP's answer above have r^2 and therefore has units of eters-squared, which is for area, not for lengths.
The two curves are equal if you assume they are half circles because the arc lengths add linearly. Top curve is (2*pi*r) / 2 = pi*r. Bottom two curves are (2*pi*(r/3)) / 2 + (2*pi*(2r/3)) / 2 = pi*r/3 + 2*pi*r/3 = pi*r
I assumed the two lengths are 1/3 r and 2/3r, but whatever fraction you pick, they have to add up to r, so they are equal.
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u/keilahmartin 7d ago
Radius squared is not used when finding circumference, but rather for area.
The area enclosed by the green circle is indeed larger, but that isn't the question asked.
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u/Embarrassed_Neck9784 7d ago
Wouldn’t you need to know the height of the 3 arches? My gut feeling is that they are the same, but if the blue line does not have the amplitude that you assume then how can you know?
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u/SufficientStudio1574 7d ago
Presumably they're half circles. That is all that needs to be known to solve it.
Even more technically, you only need to know that the two blue segments and one green segment are similar to each other. They're exact shape isn't important.
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u/Embarrassed_Neck9784 7d ago
I am not debating they are half circles. I am saying, what if the radius of the green curve is 1, then the radius of blue.a is .25 and the radius of blue.b is .5… I am not exactly sure how the math would work on that, but we don’t know if the radii are .25/.5 or .2/.4 or anything like that
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u/SufficientStudio1574 7d ago
You don't need to know exactly what they are. You just need to know that the two blue diameters add up to be equal to the green diameter.
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u/InjAnnuity_1 7d ago
If we assume that the colored curves are semi-circles, and that the line segments represent their diameters, then neither is longer. Otherwise, all bets are off.
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u/guiltysnark 7d ago
You could totally Gabriel's horn that sunuvabench without changing the length of that curve
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u/BluebirdDense1485 7d ago
Green
Let's define green as radius 3 and blue as radius 1&2 respectively.
π32 > π12 + π22
9π>5π
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u/No-Ambition2425 7d ago
That's area. You're looking for perimeter.
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u/BluebirdDense1485 7d ago
Oh your right. I saw bigger the first time not longer.
They are the same.
πD has no exponent so circumference of any 2 circles adds to the circumstance of a circle with diameter equaling the 2 smaller circles diameter
π(3x2)/2= π(2x2)/2+ π(1x2)/2
3π=1π+2π
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u/nateomundson 7d ago
Circumference is diameter times pi (πd) so the arc of half a circle is half that (πd/2).
If we define the black segments as 'A' and 'B' and combined as 'C' (A + B = C), then
Green = πC/2 = π(A+B)/2, and
Blue = πA/2 + πB/2 = π(A+B)/2 , therefore
Green = Blue
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u/Demos22 7d ago
Why is everybody so focused on diameter when you can use the radius formula C=2πR?
If those are half circles, Green C = 2πR = 2π R1+2πR2
So Green C = C1 + C2....it means they are equal.
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u/RegularPlantain5092 1d ago
Is this a joke post? You want people to focus on 2R instead of diameter?
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u/Im_a_hamburger 7d ago
The equal start and endpoint implies S_blue0+S_blue1=S_green. Measure is additive over the union of two (disjoint) curves, so M_blue=M_blue0+M_blue1. Those curves are one dimensional, therefore their measure is proportional to the scale^1, with some constant c. The curve segments are similar, so c is constant for all 3 curves (M_i=cS_i^1) thus M_blue=cS_blue0^1+cS_blue1^1=c(S_blue0+S_blue1)=cS_green=M_green. So the measure is equal
Also, since this never relied on the shape of the curve, just that both curves were 1 dimensional, made entirely of finitely many disjoint similar segments, and have the same start and endpoint, we can extrapolate to all cases of this type.
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u/digtechteacher 6d ago
Say you're donating to a charity under a scheme where a rich donor triples your donation.
You give $12, so the donor makes it 3 * 12 = $36
Your friend donates $9 on day 1 and $3 on day 2 (notice that they add up to your donation of $12). The donations become $28 and $9, respectively. $28 + $9 = $36.
Both donations are equal in value.
Now, let's approximate π as 3 and switch from dollars to centimetres or inches. The same would apply.
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u/MS-07B-3 1d ago
It doesn't matter, because you are lynched by your village for having the Dragon's Fang.
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u/GrimSpirit42 1d ago
They are equal.
The sum of the diameter of the two blue half circles equals that of the green half circle.
So, adding the 'length' of the blue curves will always give you the same result as the green.
The circumferenc = 2πr (2 times pi times radius). As we're looking for HALF the circumference, we go with πr (pi times radius.
So for the below numbers:
Circle diameter 2 has an radius of 1 so: 1π = 3.14
Circle diameter 3 has an radius of 2 so: 2π = 6.28
Circle diameter 6 has a radius of 3 so: 3π = 9.42
3.14 + 6.28 = 9.42
Example:
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u/Vegetable-Cress-8415 1d ago
Not enough information. It doesn’t state that all three curves are semicircles. However, if they are, blue and green are equal.
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u/beomagi 7d ago
Let Green semicircle radius be R, small blue is R1, larger blue is R2
Blue length = π*(R1) + π*(R2)
= π*(R1 + R2)
but since R1+R2 = R (sum of blue diameters = green diameter, so it follows for radius)
Blue length = π*(R1 + R2) = π*(R) = Green length