Find the area of the circle, length of whose circumference is equal to the sum of the lengths of the circumferences with radii 15 cm and 13 cm.

Given:

Length of circumference = Sum of the lengths of the circumferences with radii 15 cm and 13 cm

Let r be the radius of the circle.

As we know, the circumference of the circle = 2πr

So,

⇒ 2πr = 2π x 15 + 2π x 13

$⇒ 2 \times \dfrac{22}{7} \times r = 2 \times \dfrac{22}{7} \times 15 + 2 \times \dfrac{22}{7} \times 13\\[1em] ⇒ \dfrac{44}{7} \times r = \dfrac{44}{7} \times 15 + \dfrac{44}{7} \times 13\\[1em] ⇒ \dfrac{44}{7} \times r = \dfrac{660}{7} + \dfrac{572}{7}\\[1em] ⇒ \dfrac{44}{7} \times r = \dfrac{660 + 572}{7}\\[1em] ⇒ \dfrac{44}{7} \times r = \dfrac{1,232}{7}\\[1em] ⇒ r = \dfrac{1,232 \times 7}{7 \times 44}\\[1em] ⇒ r = \dfrac{1,232 \times \cancel{7}}{\cancel{7} \times 44}\\[1em] ⇒ r = \dfrac{1,232}{44}\\[1em] ⇒ r = 28 \text{ cm}$

So, radius of the circle = 28 cm

And, area of the circle = πr²

$= \dfrac{22}{7} \times 28^2\\[1em] = \dfrac{22}{7} \times 784\\[1em] = \dfrac{17,248}{7}\\[1em] = 2,464 \text{ cm}^2$

Hence, the area of the circle is 2,464 cm².