Question

Find the following ratios.

(i) The ratio of radius to circumference of the circle.

(ii) The ratio of circumference of circle with radius r to its area.

(iii) The ratio of diagonal of a square to its side, if the length of side is 7 cm.

(iv) The lengths of sides of a rectangle are 5 cm and 3.5 cm. Find the ratio of its perimeter to area.

Answer 1

(i)
Let the radius of the circle be r units.

∴ Circumference of the circle = 2$\mathrm{\pi }$r units

Radius of the circle : Circumference of the circle = r : 2$\mathrm{\pi }$r = $\frac{r}{2\mathrm{\pi }r}=\frac{{\displaystyle 1}}{{\displaystyle 2\mathrm{\pi }}}$ = 1 : 2$\mathrm{\pi }$

Thus, the ratio of radius to cirumference of the circle is 1 : 2$\mathrm{\pi }$.

(ii)
Radius of the circle = r units

∴ Circumference of the circle = 2$\mathrm{\pi }$r units

Area of the circle = $\mathrm{\pi }$r² square units

Circumference of the circle : Area of the circle = 2$\mathrm{\pi }$r : $\mathrm{\pi }$r² = $\frac{2\mathrm{\pi }r}{\mathrm{\pi }{r}^2}=\frac{2}{r}$ = 2 : r

Thus, the ratio of circumference of circle with radius r to its area is 2 : r.

(iii)
Side of the square = 7 cm

∴ Length of diagonal of the square = $\sqrt{2}$ × Side of the square = $7\sqrt{2}$ cm

Length of diagonal of the square : Side of the square = $7\sqrt{2}$ cm : 7 cm = $\frac{7\sqrt{2}}{7}=\frac{{\displaystyle \sqrt{2}}}{1}$ = $\sqrt{2}$ : 1

Thus, the ratio of diagonal of the square to its side is $\sqrt{2}$ : 1.

(iv)
Length of the rectangle, l = 5 cm

Breadth of the rectangle, b = 3.5 cm

∴ Perimeter of the rectangle = 2(l + b) = 2 × (5 + 3.5) = 2 × 8.5 = 17 cm

Area of the rectangle = l × b = 5 × 3.5 = 17.5 cm²

Perimeter of the rectangle : Area of the rectangle = 17 : 17.5 = $\frac{17}{17.5}=\frac{170}{175}=\frac{170÷5}{175÷5}=\frac{{\displaystyle 34}}{35}$ = 34 : 35

Thus, the ratio of perimeter to area of the rectangle is 34 : 35.