# Area in between two concentric circles

Area in between two concentric circles.

I am working on a problem and a bit stuck.

The problem is that we have two concentric circles, with the area inclosed by the inner circle equal to $$7000\pi$$ square meters . We are then given that the ratio of the circumference of the outer to the inner circle is equal to $$8:7$$ .

I realize that in order to solve this we need to understand the formulas for the area of a circle and the circumference of a circle.

Thus, we recognize that $$C = 2\pi r$$ and $$A = \pi r^2$$

We then conclude that $$7000\pi = \pi r^2$$ and find $$r = \sqrt{7000}$$

Now that we have this value, we need to find it in terms of the ratio $$8:7$$ in order to find the radius of the larger circle so that we can calculate its area and subtract that of the smaller circle. This is where I am stuck however, how do we convert the radius $$\sqrt{7000}$$ in to the appropriate ratio?

Do we just simply divide by seven and then multiply this value by eight?

Thanks.

• Yes, you just multiply the radius by $\frac 87$. – Ross Millikan Aug 29 '20 at 2:55
• @RossMillikan Thanks! – Ethan Aug 29 '20 at 2:56
• If the linear measure is in ratio $8:7$ then the area measure is in ratio $64:49$ ($8^2:7^2$). Which means that the ratio of the area between them to the area of the smaller circle is $15:49$. – CogitoErgoCogitoSum Aug 29 '20 at 3:05

$$7C_{outer}=8C_{inner}$$

$$7(2\pi r_{outer})=8(2\pi r_{inner})$$

$$14\pi r_{outer}=16\pi r_{inner}$$

$$r_{outer}=\frac{8}{7} \ r_{inner}$$

$$r_{outer}=\frac{8}{7}\sqrt{7000}$$

• Your first equation is incorrect . It should be $8C_{inner}=7C_{outer}$. This flows all the way through. The outer radius should be larger than the inner radius, while you have it $\frac 78$ the size. – Ross Millikan Aug 29 '20 at 2:57
• @RossMillikan Sorry about that. I realized once I read it over. Thanks for letting me know. – Kman3 Aug 29 '20 at 2:58

If you need to find the area between those two circles , use the hint given below :

$$\frac {Area_{outer}}{Area_ {inner}} =\frac{ R²_{1}} {R²_{2}}. \,$$

$$\frac{Area_{outer}} {Area_ {inner}} =\frac{ 64}{49}.$$

$$\frac{(Area_{outer} - Area_ {inner})} { Area_{inner}} = \frac{15} {49}.$$

$${Area_{outer} - Area_ {inner}} = Area_{remaining}$$

$$=\frac {15}{49}× Area_{ inner} = \frac{15}{49} × 7000π = \frac{15000π}{7} sq. unit$$

Hope it helps.

• @Ethan was that helpful ?? – A Student 4ever Aug 29 '20 at 5:57