# Two wheels of radii 2 and 6 are mutually tangent. A belt is wrapped tightly around them. What is the total length of the belt?

Jan 4, 2018

$\text{Total length} = 8 \left(\pi + \sqrt{5}\right)$

#### Explanation:

We know that the belt will cover half the circumference of both circles, and the circumference is given by $c = r \theta$. For half a circle, the angle is ${180}^{\circ} \equiv \pi \textcolor{w h i t e}{l} \text{radians}$

So, we have ${c}_{1} = 6 \pi$ and ${c}_{2} = 2 \pi$

Now, we have the section between the circles this can be found by using Pythagorus. Since we can create a right angled triangle which has a base of the two radii, $2$ and $6$, and a height equal to the difference of the two radii, $6 - 2 = 4$.

So, we have ${a}^{2} = {b}^{2} + {c}^{2}$, and $a = \sqrt{{8}^{2} + {4}^{2}} = \sqrt{80}$.

However, we have two of these, so the whole distance would be $d = {c}_{1} + {c}_{2} + 2 a = 6 \pi + 2 \pi + 2 \sqrt{80} = 8 \pi + 2 \sqrt{80} = 8 \pi + 2 \left(4 \sqrt{5}\right) = 8 \pi + 8 \sqrt{5} = 8 \left(\pi + \sqrt{5}\right)$