Can you find the length of the RED BAND wrapped around the circles? | Radii of circles are 2 and 6

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Can you find the length of the RED BAND wrapped around the circles? | Radii of circles are 2 and 6

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[bigm]

Thanks Professor. Great fun with circles!🥂❤😀

[bobapthorpe]

This explanation is very clear, very methodical, and - critically - shows multiple ways to solve for intermediate values.

[joexb]

I'm so far removed from these days but you make it so fun and understandable professor. Love when you show alternative methods.

[harikatragadda]

Smart solution!
Can also be done by extending the tangents to the x axis, forming two similar right triangles whose heights are the big and the small radius. Their Hypotenuse difference gives the length of the segment between the circles, and also the angle.

[HVAC_Tips_Tricks_Calcs]

I finally got one before watching the solution! In HVAC I use these formulae for find fan belt lengths when the two pulley diameters are known, as well as the distance from centerline to centerline of the pulleys (circles).

.

[vara]

A lengthy solution to this problem. Enjoyed watching you solving it. Keep uploading more interesting ones.

[john-ln]

35 years out of math classes I answered this. I'm a plumber. I credit my teachers. May they do well and prosper. Thank you.

[undercoveragent]

Excellent puzzle and such a clever solution. Loved it. Thank you. :)

[eoinrobson]

Thanks - always learn something new here!

[C.A.TGamingTime]

Amazing video and explanation as always ❤👍

[mohabatkhanmalak]

This formula at the end, is very usefull for determining the belt lengths for pulleys when not given or when designing pulley systems. Pulleys are usually apart, so that line OP will be longer with distance between the two pulleys 'd' added to OP. The rest is as you have shown.

[ybodoN]

OA = R and PB = r then ∠AOP = θ = arcos (R − r) / (R + r) (in radians).
So the minor arc BE is 2θ · r and the major arc AD is 2(π − θ) · R.
The tangent segment AB is 2√(rR). By symmetry, the same for DE.
Thus, the length of the red band is equal to 2(θr + 2√(rR) + (π − θ)R).

[spiderjump]

OP = 8cm
from P drop a line perpendicular to the radius of the larger circle ( that intersects with the tangent line), forming a rt triangle with 8cm hypotenuse and one side = 4 cm, this triangle is 30 60 90 rt triangle, and the other side will be 4x sqrt3
the curved part of the larger circle will be 240/360 x pi x12 = 8pi
the curved part of the smaller circle will be 120/360 xpi x4 = 4/3 pi
the straight part = 8xsqrt3
add them together
28/3pi + 8sqrt3

[northof-]

What I did was this:
Imagine you cut the band at each touch points on the large circle and wrap it around the big circle,
giving you a 2πx6.
That leaves approximately half of the small circle covered with the red band = πx2
My result is 43.9823 so not that far off as a quick approximation

[vidyadharjoshi]

Let's label the points on big circle as A & B, small circle as C & D. Draw a perpendicular from P to OA meeting at E. OE = 4, OP = 8 so AC = PE = 6.93
So Angle EPO = 30, So the angles of the Red arcs of Big and small will be 240 deg &120 deg.
Length of red line = (240/360*(2*pi*6) + (120/360)*(2*pi*2) +2*6.93 = 25.13 + 4.18 + 13.86 = 43.17

[HappyFamilyOnline]

Great presentation 👍👍
Thanks for sharing😊

[d.l.harrington]

While using PreMath was excellent (at least until my mind got twisted) here is the formular I used when replacing v-belts when the size was not evident. 2C + [1.57 x (D1 + D2)] + 1-in. Probably not as accurate but close enough for my work. I did have some leeway as there was adjustments to be made. Try it sometime if you ever need a belt. Does not account for the groves.

[MichaelWillems]

Very good.
Once you come up with the parallelograms, this is very simple. The question is: how do you come up with the parallelograms? That is not obvious (until you see it)!

[JSSTyger]

Got it right on those nose. Great as always.

[MrPaulc]

(4/3)pi + 12*(2/3)pi is the length that touches the circles.
(4/3)pi + 8pi = (28/3)pi for the touching part.
Distance between centres is 8. To make right triangle, use a point that is 4 away from large circle's centre.
8^2 - 4^2 = 48, so sqrt(48) which corresponds to each non-touching length.
2*sqrt(48) = 8*sqrt(3)
Length of red band is (28/3)pi + 8*sqrt(3).
If you want decimal, that approximates to 29.3216 + 13.856...
29.32 + 13.86 should be close enough.
43.18 ish.