The ratio between the diameters of two circles is 3 : 5. Find the ratio between their :

(i) radii

(ii) circumferences

(iii) areas

(i) The ratio between the diameters of two circles = 3 : 5

Let the diameter of the 1st circle be 3a and that diameter of the 2nd circle be 5a.

Radius of the circle = $\dfrac{\text{Diameter}}{2}$

Radius of 1st circle = $\dfrac{3a}{2}$

Radius of 2nd circle = $\dfrac{5a}{2}$

Now, the ratio of their radii:

= $\dfrac{\dfrac{3a}{2}}{\dfrac{5a}{2}}$

= $\dfrac{3a}{5a}$

= $\dfrac{3}{5}$

Hence, the ratio of the radii is 3:5.

(ii) The circumference of a circle = 2πr

Circumference of 1st circle = 2π x $\dfrac{3a}{2}$

Circumference of 1st circle = 2π x $\dfrac{5a}{2}$

Now, the ratio of their circumferences:

= $\dfrac{2π \times \dfrac{3a}{2}}{2π \times \dfrac{5a}{2}}$

= $\dfrac{3a}{5a}$

= $\dfrac{3}{5}$

Hence, the ratio of the circumferences is 3:5.

(iii) Area of the circle = πr²

Area of 1st circle = π x $\Big(\dfrac{3a}{2}\Big)^2$ = π x $\Big(\dfrac{9a^2}{4}\Big)$

Area of 2nd circle = π x $\Big(\dfrac{5a}{2}\Big)^2$ = π x $\Big(\dfrac{25a^2}{4}\Big)$

Now, the ratio of their areas:

= $\dfrac{π \times \dfrac{9a^2}{4}}{π \times \dfrac{25a^2}{4}}$

= $\dfrac{9a^2}{25a^2}$

= $\dfrac{9}{25}$

Hence, the ratio of the areas is 9:25.