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Areas Related to Circles — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsAreas Related to Circles3 Marks
Question
An archery target has three regions formed by the concentric circles as shown in the figure. If the diameters of the concentric circles are in the ratio 1:2:3, then find the ratio of the areas of three regions.
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Answer

Let the diameters of concentric circles be k,
$\therefore$ Radius of concentric circles are $\frac{\text{k}}{2},\text{k}\text{ and }\frac{3\text{k}}{2}.$
$\therefore$ Area of inner circle , $\text{A}_1=\pi\Big(\frac{\text{k}}{2}\Big)^2=\frac{\text{k}^2\pi}{4}$
$\therefore$ Area of middle region,
$\text{A}_2=\pi(\text{k})^2-\frac{\text{k}^2\pi}{4}=\frac{3\text{k}^2\pi}{4}$
$[\therefore\text{area of ring}=\pi(\text{R}^2-\text{r}^2),$ where R is radius of outer and r is radius of inner ring]
and area of outer region, $\text{A}_3=\pi\Big(\frac{3\text{k}}{2}\Big)-\pi\text{k}^2$
$=\frac{9\pi\text{k}^2}{4}-\pi\text{k}^2=\frac{5\pi\text{k}^2}{4}$
$\therefore\text{Required ratio}=\text{A}_1:\text{A}_2:\text{A}_3$
$=\frac{\text{k}^2\pi}{4}:\frac{3\text{k}^2\pi}{4}:\frac{5\pi\text{k}^2}{4}=1:3:5$

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